SHOCK WAVES IN FLOW OF AN INCOMPRESSIBLE LIQUID IN COLLAPSING PIPES: APPLICATION TO LARGE BLOOD VESSELS

8. L. I. Sedov, Mechanics of a Continuous Medium, Vol. i [in Russian], Nauka, Moscow (1983).
SHOCK WAVES IN FLOW OF AN INCOMPRESSIBLE LIQUID IN COLLAPSING
PIPES: APPLICATION TO LARGE BLOOD VESSELS
Yu. Z. Saakyan UDC 532.542
Flow of a liquid in collapsing pipes is of great interest for problems in the
mechanics of blood circulation, since collapse can take place in many blood
vessels. This effect forms the basis for a large number of diagnostic and
therapeutic methods, and also for methods of investigating the system of
blood circulation. Consequently the mechanics of collapsing pipes has been
studied intensively of late [I], but the available studies are far from
exhausting the theoretical or the applied aspects of the problem. This applies
also to the study of discontinuous solutions such as shock waves which
describe steep fronts of opening or narrowing of a blood vessel~ The most
studied phenomenon is unsteady flow caused by change in the external pressure
[2]. There is an explanation in [3-6] of the effect on the process of
formation of discontinuities in collapsing pipes due to such factors as
friction on the wall, distributed lateral outflow, the presence of a stagnant
zone in the flow, and viscoelasticity of the wall. The origin of some acoustic
phenomena in the arteries is connected by some with the propagation of discontinuities;
these phenomena include Korotkov sounds, used in the determination
of the arterial pressure of blood [i, 7]~ The present study considers quasione-
dimensional flow of a viscous incompressible liquid in a collapsing pipe
of finite length and made of a nonlinear viscoelastic material; there is a
study of the conditions in which discontinuities arise in such systems, and
an investigation of the structure of shock waves with allowance for the
effect of the surrounding tissues.
i. The equations of motion of a
cross section area A and perimeter F
liquid with constant density p along a pipe with
will be written in the form
aA+_a (1.1)
at Ox
Ou Oa I Op
P "~'+u'-~x/+--=-qDox (1o2)
Here x is the axial coordinate (0 ~ x ~ L, L is the length of the pipe), ~=(F~JA);
u, p are the mean axial velocity and pressure over the cross section, ~ is the specific
outflow of the liquid across the lateral surface of the pipe, and ~w is the mean viscous
tangential stress on it over the perimeter.
We assume, in accordance with [8], that the dependence of ~ on the parameters of the
problem is of the form ~ = a(p -- Pc) + 7uA, where Pc is the pressure at the outlet of the
pipe, and ~ > 0 and 7 > 0 are constant coefficients. For ~w we postulate a linear dependence
on u, corresponding to an instantaneous Poiseuille velocity profile for the flow
of a liquid with a dynamic coefficient of viscosity ~ along a pipe of elliptical cross
section: q=4xB(6+|/6)(u/A). 6(F, A) being the ratio of the minor and major axes of the
ellipse.
Equations (iol) and (1.2) must be supplemented by a relationship connecting the pressure
p with the cross section area A and reflecting the theological properties and nature
of the motion of the pipe and the medium surrounding it. We shall assume that the pipe
is located in an incompressible anisotropic continuous medium of density Pm, occupying
in the nondeformed state the area between coaxial cylinders of radii R 0 and r 0 (r 0 is at
the same time the nondeformed radius of the mean surface of the shell modeling the pipe).
We assume, moreover, that on a certain segment [i < x < s an external pressure pe(X) is
Moscow. Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza,
No. 6, pp. 44,50, November-December, i987. Original article submitted March 6, 1987.
862 0015-4628/87/2206-0862512.50 9 1988 Plenum Publishing CorpOration

Peripheral arterial blood pressure monitoring adequately tracks central arterial blood pressure in critically ill patients: an observational study

Open Access
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Vol 10 No 2 Research
Peripheral arterial blood pressure monitoring adequately tracks
central arterial blood pressure in critically ill patients: an
observational study
Mariano Alejandro Mignini1, Enrique Piacentini1,2 and Arnaldo Dubin3
1Critical Care Unit, Clínica Bazterrica, Buenos Aires, Argentina
2Critical Care Unit, Hospital Mutua Terrassa, Terrassa, Spain
3Critical Care Unit, Sanatorio Otamendi y Miroli, Buenos Aires, Argentina
Corresponding author: Arnaldo Dubin, arnaldodubin@speedy.com.ar
Received: 25 Oct 2005 Revisions requested: 19 Dec 2005 Revisions received: 2 Jan 2006 Accepted: 13 Feb 2006 Published: 8 Mar 2006
Critical Care 2006, 10:R43 (doi:10.1186/cc4852)
This article is online at: http://ccforum.com/content/10/2/R43
© 2006 Mignini et al.; licensee BioMed Central Ltd.
This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/2.0),
which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
Abstract
Introduction Invasive arterial blood pressure monitoring is a
common practice in intensive care units (ICUs). Accuracy of
invasive blood pressure monitoring is crucial in evaluating the
cardiocirculatory system and adjusting drug therapy for
hemodynamic support. However, the best site for catheter
insertion is controversial. Lack of definitive information in
critically ill patients makes it difficult to establish guidelines for
daily practice in intensive care. We hypothesize that peripheral
and central mean arterial blood pressures are interchangeable in
critically ill patients.
Methods This is a prospective, observational study carried out
in a surgical-medical ICU in a teaching hospital. Fifty-five
critically ill patients with clinical indication of invasive arterial
pressure monitoring were included in the study. No interventions
were made. Simultaneous measurements were registered in
central (femoral) and peripheral (radial) arteries. Bias and
precision between both measurements were calculated with
Bland-Altman analysis for the whole group. Bias and precision
were compared between patients receiving high doses of
vasoactive drugs (norepinephrine or epinephrine >0.1 μg/kg/
minute or dopamine >10 μg/kg/minute) and those receiving low
doses (norepinephrine or epinephrine <0.1 μg/kg/minute or
dopamine <10 μg/kg/minute).
Results Central mean arterial pressure was 3 ± 4 mmHg higher
than peripheral mean arterial pressure for the whole population
and there were no differences between groups (3 ± 4 mmHg for
both groups).
Conclusion Measurement of mean arterial blood pressure in
radial or femoral arteries is clinically interchangeable. It is not
mandatory to cannulate the femoral artery, even in critically ill
patients receiving high doses of vasoactive drugs.
Introduction
Invasive arterial blood pressure monitoring is a common practice
in intensive care units (ICUs). The most frequent indication
for invasive arterial blood pressure monitoring is for continuous
measurement in hemodynamically unstable patients [1].
The radial artery is most commonly used, with the femoral
artery being the second choice. One or the other is used in
92% of arterial cannulations [2]. Accuracy of invasive blood
pressure monitoring is crucial in evaluating the cardiocirculatory
system and adjusting drug therapy for hemodynamic support.
However, the best site for catheter insertion is
controversial. For some clinicians, the femoral artery is the preferred
site because of its lower rate of mechanical (occlusion,
accidental loss, thrombosis) and infectious complications [2-
4]. The accuracy of peripheral blood pressure compared with
central blood pressure measurements has been evaluated by
many authors in patients undergoing cardiac surgery [5-12].
Unfortunately, in this setting the population is homogeneous
and very different from critically ill patients found in a medical
and surgical ICU.
In critically ill patients treated with vasoactive drugs, Dorman
and colleagues [13] reported that radial arterial pressure monitoring
significantly underestimates central arterial pressure.
Insertion of a femoral line allowed a substantial reduction of
the infusion rate of vasoactive drugs in these patients [13].
ICU = intensive care unit; SD = standard deviation.
Critical Care Vol 10 No 2 Mignini et al.
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These findings might imply that femoral placement of arterial
lines is the gold standard for invasive arterial blood pressure
monitoring in shock patients. Nevertheless, that study involved
a selected group of patients with postoperative septic shock
and only norepinephrine was used as a vasoactive drug. In
addition, interchangeability between measurements was not
adequately evaluated.
Lack of definitive information in critically ill patients makes it difficult
to establish guidelines for daily practice in intensive care.
We hypothesize that peripheral and central mean arterial
blood pressures are interchangeable in critically ill patients. To
test our hypothesis we compare simultaneous measurements
of arterial blood pressure in peripheral and central arteries in a
heterogeneous population of critically ill patients using formal
Bland-Altman analysis [14].
Materials and methods
Study population
The study was approved by the Hospital Ethics Committee
and the need for informed consent was waived because no
additional procedures apart from usual intensive care practice
were involved.
Fifty-five critically ill patients admitted to our mixed (medicalsurgery)
ICU from 16 December 1999 to 22 December 2000
were studied. Inclusion criteria were: clinical indication of invasive
arterial pressure monitoring, such as cardiovascular instability,
use of intravenous vasoactive agents, and need for
frequent sampling of arterial blood [1]; and the need to change
the insertion site of the arterial line. Fever and suspicion of
catheter-related infection were the main reasons to change
the arterial insertion site. The indication was determined following
internationally accepted guidelines [15]. Exclusion criteria
were: post-cardiac surgery patients; patients with
catheter malfunctioning detected by the 'fast flush test' (the
pressure in the line was rapidly increased to 300 mmHg by
flushing the system with the continuous flow mechanism and
the resulting waveform was analyzed to determine the
response of the system; ideally, one large and one small oscillation
should occur, after which the waveform should be
returned to the baseline [16]); patients who needed to be in
positions other than the semirecumbent supine; patients with
clinical history of peripheral arterial occlusive disease. These
two last criteria were based on the possibility of registering
artificially or pathologically modified data. Post-cardiac surgery
patients were excluded because they are a homogeneous
population in which the issue of radial-to-femoral arterial pressure
gradient has been well investigated [5-12].
The clinical features of each patient's disease guided indications
of invasive arterial blood pressure monitoring and all
patients received standard treatments following the guidelines
for the pathologies diagnosed or suspected.
Patients were separated into two groups: those receiving high
doses of vasoactive drugs (dopamine ≥10 μg/kg/minute or
epinephrine or norepinephrine ≥0.10 μg/kg/minute); and
those receiving low doses of vasoactive drugs (dopamine <10
μg/kg/minute or epinephrine or norepinephrine <0.10 μg/kg/
minute) or no vasoactive. Demographic data (sex and age),
APACHE II score [17] at enrollment, number of organ failures
(SOFA score) [18], and type (dopamine, epinephrine or norepinephrine)
and dose of vasoactive drugs used were
recorded.
Study design
For femoral arteries, 14- or 16-gauge catheters were used
(Secalon T 16 G/2.0 × 160 mm Ohmeda, Swindon, Great
Britain) and for radial arteries, a 20-gauge catheter was used
(Vasculon 2 20 G/32 mm BOC Ohmeda, AB SE-2506, Helsingborg,
Sweden). The catheters were inserted with their tips
pointing towards the blood flow. Indwelling devices were connected
to a continuous-flush transducer system through a
rigid plastic tube measuring 120 cm in length in all cases,
regardless of whether central or peripheral arteries were used
(Becton Dickinson DTX PLUS DT 4812, BD Infusion Therapy
Systems, Inc., Sandy, Utah, USA). Both transducers were
placed at the same level (right atrium) on a plastic support and
zeroed to atmosphere. The arterial blood pressure signals
were recorded and displayed on a bedside monitor (Viridia
M1205A 24CT, Hewlett Packard, Andover, MA, USA) and the
waveforms were simultaneously and permanently registered
online. The whole tubing system was flushed with sterile normal
saline to eliminate air bubbles and tested for system loss
(for instance any kind of fluid leak from the circuit). The monitoring
device was connected to a permanent pressurized
washing system. Curve characteristics were constantly evaluated
using a rapid flush test to rule out occlusion or catheter
malposition [16].
The readings over the first five minutes after the insertion of the
second catheter were simultaneously recorded, and the mean
values for systolic, diastolic, and mean arterial pressures were
calculated for both catheters. The data from the entire population
were analyzed to determine the global accuracy of the
peripheral measurement of mean blood pressure. Next, the
two groups were analyzed separately and differences
between groups were evaluated to determine the interchangeability
of peripheral and central mean arterial blood measurements.
We focused the analysis of interchangeability on mean
arterial pressure because the tissue perfusion pressure is
mainly given by mean arterial pressure rather than by systolic
or diastolic pressure.
Statistical analysis
Data were analyzed using the Bland and Altman method [14].
Bias, precision, and 95% limits of agreement of the simultaneous
measurements were calculated [14]. Bias and precision
between groups were compared using unpaired t tests.
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Descriptive data are expressed as mean ± standard deviation
(SD). Statistical significance was defined as p < 0.05.
Results
The characteristics of the 55 patients are shown in Table 1.
The most common reasons for admission were respiratory
insufficiency, shock, and postoperative monitoring. Forty
patients were classified as receiving high doses of vasoactive
drugs and 15 were considered to be receiving low doses.
Patients in the high-dose group were receiving dopamine (n =
12, doses ranging from 11 to 46 μg/kg/minute), norepinephrine
(n = 16, doses ranging from 0.11 to 13.5 μg/kg/
minute), or epinephrine (n = 12, doses ranging from 0.33 to
7.4 μg/kg/minute); 5 patients in this group were also receiving
dobutamine simultaneously. Five patients in the low-dose
group were receiving dopamine in doses ranging from 3 to 7
μg/kg/minute, one patient was on norepinephrine 0.063 μg/
kg/minute, and nine were not receiving any vasoactive drug.
No differences were found in systolic, diastolic, or mean arterial
blood pressure measured in the femoral artery versus the
radial artery in the entire population or in either of the two
groups (Table 2).
For the whole population, bias (mean difference between
simultaneous measurements) ± precision (SD of the difference
between those values) of simultaneous femoral and
peripheral mean arterial blood pressure measurements was 3
± 4 mmHg. With these values, the 95% limits of agreement
(mean ± 2 SD of the difference between simultaneous measurements)
are 16 mmHg (Figure 1). No differences in bias ±
precision were found between the high (3 ± 4 mmHg) and
low-dose (3 ± 4 mmHg) groups.
Discussion
The main finding of this study is that central and peripheral
mean arterial blood pressures appear to be interchangeable.
The 95% limits of agreement of 16 mmHg is not a clinically relevant
difference in mean arterial pressure and the two measurements
agree regardless of whether patients were receiving
vasoactive drugs.
O'Rouke and colleagues [19] have shown that there are no differences
in mean arterial blood pressure simultaneously measured
in the aorta and radial arteries in healthy volunteers.
However, systolic and diastolic arterial blood pressures are
higher and lower, respectively, in radial arteries than in the
aorta. This phenomenon is known as distal pulse amplification
and is due to the characteristics of the vascular tree. Briefly, a
pulse waveform entering the aorta is exposed to a sudden
impedance change at the capillary level, resulting in a large
increment in resistance and producing reflected pulse waveforms.
Those waves are added to the following ones, producing
higher peaks than the original aortic systolic peak at
different distances from the aortic origin. This distal pulse
amplification is always present when peripheral vascular
resistance is high [19]. In our study we found no evidence of
this phenomenon. In fact, systolic, mean, and diastolic pressures
were higher in the femoral artery than in the radial artery.
Lack of physiological distal pulse wave formation could be due
to a vasoactive effect in shock patients. Thus, although
vasoactive drugs act mainly on resistance vessels, they also
affect conductance vessels, which could alter peripheral arterial
blood pressure measurements.
Yazigi and colleagues [20] studied normal volunteers to determine
whether radial arterial pressure accurately reflects
Table 1
Characteristics of the study population (n = 55)
Characteristic Vasoactive dose
High (n = 40) Low (n = 15)
Gender (men:women) 19:21 7:8
Age (years) 68 ± 16 66 ± 15
APACHE II score 24 ± 7 16 ± 7
Number of organ failures 4 ± 2 2 ± 1
Patients on mechanical ventilation
(%)
40 (100) 15 (100)
Patients with pulmonary artery
catheter (%)
32 (80) 3 (20)
Patients with vasoactive drugs (%) 40 (100) 7 (47)
Patients with dopamine (%) 12 (30) 6 (40)
Patients with epinephrine (%) 12 (30) 0 (0)
Patients with norepinephrine (%) 16 (40) 1 (7)
Figure 1
Plot of differences against a fPelmoto oraf ld aifnfedr erandcieasl mageainn satr taevreiarla bgleoso do fp sriemsuslutarenseous measurements of
femoral and radial mean arterial blood pressures. The solid line represents
bias (the mean difference between simultaneous measurements).
Dotted lines show 95% limits of agreement (bias ± 1.96 standard deviation).
The small bias (mean difference between simultaneous measurements)
and the narrow 95% limits of agreement suggest the
interchangeability of both measurements.
Critical Care Vol 10 No 2 Mignini et al.
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changes in blood pressure induced by nicardipine. They concluded
that peripheral arterial pressure is an accurate measure
of central arterial pressure in this setting, and they found no
distal pulse amplification.
Invasive arterial blood pressure measurement is a common
practice during shock management in the ICU, and the radial
artery is the most common site of insertion, followed by the
femoral artery. Given the large number of patients requiring
high doses of vasoactive drugs during relatively prolonged
periods of time and the need to change arterial lines to avoid
infectious complications [21-23], it is important to determine
whether the measurements are equivalent in alternative cannulation
sites.
To our knowledge, this issue has been systematically
approached only by Dorman and colleagues [13] in 14 postoperative
patients with septic shock receiving high doses of
norepinephrine (86 ± 25 μg/minute). A systematic underestimation
of mean and systolic arterial blood pressure was found
for measurements in the radial artery with respect to the femoral
artery. Consequently, this finding allowed the doses of
norepinephrine to be decreased and even withdrawn in two
patients. After changing the dosage of norepinephrine, differences
between mean radial and femoral arterial blood pressures
disappeared [13].
Discrepancies between our results and those reported by Dorman
and colleagues might be related to different issues. First,
there are probably intrinsic differences in the populations studied.
The diagnoses of our patients were more heterogeneous
(medical and postoperative patients with and without shock)
and a broader range of doses of different vasoactive drugs
was used. Only 17 (16 in the high dose group and one in the
low dose group) patients included in our study were receiving
norepinephrine; however, bias and precision between peripheral
and central arterial blood pressure was the same in the different
groups.
Another source of discrepancy might be the measurement
technique used. We tried to minimize variability in the measurement
system. For this reason, simultaneous recordings of
both pressures were registered on the same monitor using
transducers, plastic lines, and washing systems sharing similar
features. Nevertheless, the size of the catheters inserted at different
sites was different in our study. Although the intravascular
portion of the catheter has minimal effect on the accuracy
of measurement [16], we cannot rule out the possibility that
the pulse wave might be modified by different cannula sizes.
Our results might be biased by measurements through smaller
catheters in peripheral arteries. However, the small bias found
in this study suggests that our results were not influenced by
this issue.
Finally, Dorman and colleagues used t tests to compare radial
and femoral arterial blood pressure measurements; however,
when the main issue to be addressed is agreement between
different measurements of a variable, the best statistical
approach is the Bland and Altman method [14]. There is no
definition of the extent to which differences between both
measurements might be relevant. Bland and Altman suggested
that if the value of the 95% limits of agreement of two
methods is not clinically important, they might be interchangeable
[14]. The small bias and its narrow standard deviation
between peripheral and central arterial blood pressure measurements
suggest their interchangeability.
Conclusion
In this study, peripheral and central measurements of arterial
blood pressure showed good agreement regardless of
vasoactive drug use. Our results suggest that these two measurements
are interchangeable and it is, therefore, not mandatory
to cannulate the femoral artery to measure arterial blood
pressure, even in critically ill patients receiving high doses of
vasoactive drugs.
Competing interests
The authors declare that they have no competing interests.
Key messages
• Femoral and radial mean arterial blood pressures
showed good agreement regardless of the use of
vasoactive drugs.
• Our results suggest that these two measurements are
interchangeable.
Table 2
Mean, systolic and diastolic arterial pressures in both groups
MAP (central) MAP (peripheral) SAP (central) SAP (peripheral) DAP (central) DAP (peripheral)
Overall (n = 55) 85 ± 17 82 ± 17 135 ± 31 126 ± 30 63 ± 14 62 ± 13
High dose (n = 40) 85 ± 16 82 ± 15 137 ± 31 124 ± 28 63 ± 13 62 ± 12
Low dose (n = 15) 84 ± 20 81 ± 20 130 ± 31 130 ± 33 62 ± 19 60 ± 16
All values are mean ± standard deviation. Overall, entire study population; High dose, high dose vasoactive drug group; Low dose, low dose
vasoactive drug group. DAP, diastolic arterial pressure MAP, mean arterial pressure; SAP, systolic arterial pressure.
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Authors' contributions
MAM and EAP participated in the conception and design of
the study, in the acquisition analysis and interpretation of the
data and drafted the manuscript. AD participated in the conception
and design of the study, in the analysis and interpretation
of the data, revised the manuscript critically for important
intellectual content and gave final approval of the version to be
published.
Acknowledgements
This study was solely funded by the Department of Intensive Care,
Clínica Bazterrica.
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High-Energy Shock Waves Induce Blood Flow Reduction in Tbmors1

[CANCER RESEARCH 53. 1590-1595. April I. 19931
High-Energy Shock Waves Induce Blood Flow Reduction in Tbmors1
Fernando Gamarra, Fritz Spelsberg, Gerhard E. H. Kuhnle, and Alwin E. Goetz2
Institute of Surgical Research IF. G.. F. S.. G. E. H. K.j and Institute of Anesthesiology ¡A.E. G.j. Luilwif>~Ma.*iniilians-llni\:ersit\ Munich, Klinikum Griisshtidem. Munich.
Germani
ABSTRACT
We have studied the effect of extracorporeally applied high-energy
shock waves (HESW) on blood flow in amelanotic melanomas (A-Mel-3).
Two tumors were implanted in the dorsal skin of 21 Syrian golden ham
sters. One of the tumors was treated with 200 HESW, and the other served
as an intraindividual control. Mean blood flow in the whole tumor, or the
tumor excluding necrotic areas, was quantitatively measured using autoradiography
with iodo['4C]antipyrine at 30 min (n = 5), l h (n = 5), 3 h
(n = 5), and 12 h (n = 6) after HESW treatment. As measured for the
whole tumor, blood flow in the controls was 23.4 ±7.9 ml/100 g/min
(median ±SE) and thus in the range reported in the literature. Thirty min
or l h after the application of HESW, tumor perfusion was reduced to 6
±4% or 5 ±4% (median ±SE) of the corresponding controls, respec
tively. Three h after treatment, perfusion increased slightly to 7 5% and
after 12 h increased significantly to 55 ±25% of the corresponding
controls. Values measured excluding the necrotic areas were higher in all
groups. Temporary reduction of tumor perfusion after treatment with
HESW was interpreted as a consequence of HESW-induced damage to
tumor microcirculation. These effects should be taken into account for
maximizing the therapeutic efficiency of HESW on tumors and for com
bining HESW treatment with other therapeutical modalities.
INTRODUCTION
Extracorporeally generated HESW1 have become the standard
treatment for fragmentation of kidney stones ( 1) and have been ap
plied for lithotripsy of gallstones (2). This was made possible because
shock waves can be focused on targets within the body with a mini
mized effect on the tissue surrounding those targets.
The use of HESW as a means for nonsurgical, local tumor treatment
has also been suggested (3). First investigations have been carried out
demonstrating the cytotoxic effects of HESW on tumor cells in vitro
(3, 4). Furthermore, treatment of experimental tumors in vivo with
HESW has been shown to induce the delay of tumor growth (3, 5, 6).
HESW are known to have damaging effects on the vasculature of
normal tissues; side effects of shock wave lithotripsy of renal stones
or gallstones include edema formation, hemorrhage, and reduction of
tissue perfusion (7, 8). Damage to microcirculation has been con
firmed by histology, electron microscopy, and intravital microscopy
(9, 10).
Similar effects of HESW have been observed on tumor vasculature
(11, 12). This is of particular interest if HESW are tobe used for tumor
therapy. HESW-induced damage of tumor vasculature with subse
quent impairment of perfusion would contribute to tumor cell death,
in addition to the direct cytotoxic effects of the shock wave itself. Cell
death secondary to perfusion defects would depend on their extent and
duration. Such mechanisms of action have been elucidated for mo
dalities of tumor therapy like hyperthermia (13) or photodynamic
therapy (14, 15). Moreover, perfusion changes after HESW have to be
Received 10/16/92: accepted 1/25/93.
The costs of publication of this article were defrayed in part by the payment of page
charges. This article must therefore be hereby marked advertisement in accordance with
18 U.S.C. Section 1734 solely to indicate this fact.
' Supported by grants of the Bundesministerium fürForschung und Technologie
(0706903 A5) and the Kurt KörberStiftung.
2 To whom requests for reprints should be addressed, at Institute of Anesthesiology,
Klinikum Grosshadern. Ludwig-Maximilians-University. Marchioninistrasse 15, 8000
Munich 70. Germany.
' The abbreviations used are: HESW, high-energy shock waves; IAP. 4-iodo|iV-mi-r/iv/-
l4C)antipyrine; MAP. mean arterial blood pressure: ROI. region of interest.
taken into account for therapeutic approaches combining HESW and
chemotherapy or other treatment modalities.
The aim of this study was therefore to quantify the extent and
duration of perfusion changes in tumors after the application of
HESW.
MATERIALS AND METHODS
Animals and Tumors. The experiments were performed in 21 male Syrian
golden hamsters (70-80 g) bearing two amelanotic hamster melanomas (AMel-
3) ( 16) in the dorsal skin. During tumor implantation. HESW application,
and blood flow measurements the animals were anesthetized with pentobarbital
(Nembutal; Sanoft-Ceva. Hannover. Germany; 60 mg/kg i.p.). For tumor im
plantation the dorsal skin was shaved and chemically depilated (Pilcamed;
Schwarzkopf GmbH. Lübeck.Germany). About 5 x IO6 A-Mel-3 cells were
inoculated s.c. at two paravertebral sites (thoracic and lumbar regions) in the
dorsal skin. Seven days after implantation tumors had grown to diameters of
7-10 mm, corresponding to volumes of 160-250 mm1 [volumes were calcu
lated as described by Weiss et al. (5)1.
HESW Application. HESW were electrohydraulically generated with the
Dornier lithotripter model XL1 (Dornier Medizintechnik. Germering. Germa
ny), as described previously (1). Briefly, an underwater spark discharge pro
duces a radially expanding shock wave which is reflected on a metal semiellipsoid
(Fig. 1). Thus the shock wave is concentrated at a focal site where
maximum pressures of 800 bar are reached within I ns. The 50% isobar of the
pressure field is currently defined as the shock wave focus; it has a diameter
of 5 mm and measures 22 mm in the longitudinal axis (17).
At day 7 of tumor growth the animals were placed into plexiglas tubes
which were covered with 2 mm of styrotbam on the inner side to protect the
body from the high-pressure field of the shock waves. The tumor-hearing
dorsal skin was elevated through a slit in the tube, fixed with three sutures
along a plastic arc (Fig. 1). and sealed watertight with a surgical incision drape
(Opraflex; Lohmann GmbH, Neuwied. Germany) shielding the space between
the skin and the tube. This arrangement made it possible to submerge the
tumors under water (water bath temperature. <36°C) while the rest of the
animals remained dry inside the tube.
One of the tumors was randomly chosen for HESW treatment. It was
positioned at the shock wave focus, which was localized by two low-energy
laser beams intersecting there. Two hundred HESW were applied on the tumor
at a fixed discharge voltage of 15 kV, a condenser capacity of 80 nF, and a
HESW application frequency of 2.3 Hz. Thus the overall treatment time was
87 s. The second tumor, located at a distance of about 30 mm from the first one,
was not exposed to HESW (Fig. 1).
Measurement of Tumor Blood Flow. Tumor blood flow was measured
with the autoradiographic tissue equilibration technique developed by Kety
(19) and Sakurada (18). Polyethylene catheters (Portex. Ltd., Hythe, Kent,
England) were implanted into the right carotid artery (outer diameter, 0.96 mm;
inner diameter. 0.58 mm), femoral artery, and superior vena cava (outer di
ameter. 0.61 mm: inner diameter. 0.28 mm). Forty jiCi of the inert, readily
diffusible compound IAP (NEN Research Products. Du Pont de Nemours.
Dreieich, Germany) were evaporated to dryness and redissolved in 0.5 ml 0.9%
NaCI solution. The carotid catheter was cut at a length of 35 mm. and two
arterial blood samples of approximately 20 (jl were drawn into heparinized
glass capillaries. The IAP solution was then injected through the superior vena
cava catheter by means of an infusion pump (Harvard Appliance. Ltd., Kent,
England) with a constant flow over 30 s. During the infusion period, further
arterial blood samples (approximately 20 ul each) were withdrawn from the
freely flowing carotid catheter every 2-3 s. MAP in the femoral artery was
registered continuously during the experiment. Exactly 30 s after the start of
the IAP infusion both tumors were rapidly resected, immediately frozen in
liquid nitrogen, and stored at -70°C.
1590
TUMOR PERFUSION AFTER SHOCK WAVES
lasor
semiallipsoidal
reflector
spark gap
Fig. 1. Experimental setup for application of HESW under water. The animals were
placed in a plexiglas tube. The tumor-bearing back skin was extended through a slit in the
tube. After randomization, one of the tumors was positioned in the shock wave focus with
the help of two low-energy laser beams crossing there. The second tumor was beyond the
focus area and served as an untreated control. HESW were electrohydraulically generated
and fix-used after reflection on a metal semiellipsoid. Water flowing into the tube was
continuously evacuated by a suction device.
Arterial blood samples were weighed and mixed with scintillation fluid to
determine MC activity with a beta counter (Rack Beta 1219; LKB Wallac.
Turku, Finland) and calculate the 14C concentration in the blood.
The MC concentration in tissue was visualized autoradiographically. The
tumor was cut alternately into 20-um and 7-um cryosections (cryostat at
-20°C): a 20-um section for autoradiography was followed by a 7-um section
for histology and a 100-um slice, which was discarded. Twenty-um sections
were placed on X-ray film (NMC: Eastman Kodak. Rochester, NY) for 2
weeks together with calibrated UC tissue standards (I4C microscales; Amersham
Buchler GmbH. Braunschweig. Germany). Seven-urn sections were
stained with hematoxylin and eosin.
The autoradiograms showing tissue I4C distribution in tumor sections were
evaluated densitometrically with an image analysis system (IPS Autoradiog
raphy Software Package; Kontron GmbH. Eching. Germany). Images of the
transilluminated autoradiograms of tumor sections and the corresponding UC
tissue standards were acquired with a CCD videocamera (XC-77; Sony. Co
logne, Germany) coupled to a macroviewer, digitized, and displayed on a
monitor. The images were compared to the corresponding hematoxylin and
eosin-stained histology. By means of a digitizer table. ROIs were selected as
follows: ROI I included the whole cross-section of the tumor without the
surrounding normal tissue, and ROI 2 included the cross-section of the tumor
without the surrounding normal tissue and without necrotic areas in the tumor.
The demarcation of the ROIs was performed by comparing the autoradiograms
with histology. The limits between normal and tumor tissue were easy to
establish. Tumor necrosis was identified as an area with no cellular structures
in hematoxylin and eosin histology.
Average blood flow within each ROI was calculated by an iterative poly
nomial regression with a computer program integrated into the image analysis
system (IPS Autoradiography Software Package) according to Kety's equation
(18):
c/m =AX C,(t) X , 'dt
where C,( T) is the tissue concentration of IAP at the end of the infusion period
(T = 30 s). Ca(f ) is the arterial concentration at time t after beginning with the
infusion of IAP. A is the blood tissue partition coefficient of IAP in A-Mel-3
tumors, and A"is a parameter which is related to blood flow F as follows:
F = K X A/m
where m is a value between 0 and I defining the extent to which IAP diffusional
equilibrium is established between tissue and blood. We assumed no
diffusion barriers for IAP between tumor vessels and interstitial space and
therefore chose m = 1.
Twenty autoradiograms corresponding to 20 levels of each single tumor
were evaluated, and the mean blood flow value of each tumor was calculated.
The blood tissue partition coefficient A was determined in separate exper
iments. Two hamsters, each bearing three tumors in the dorsal skin, were
tracheotomized and artificially ventilated with a 70"7r N2O/30% O; mixture.
Anesthesia was maintained by 1.5% enflurane. Catheters were placed in the
right carotic and femora] arteries and superior vena cava. To avoid metabolic
degradation of IAP during the time needed for equilibration between blood and
tissue, laparotomy was performed and both renal arteries and veins, the hepatic
portal vein, and hepatic artery were ligated (18. 20). Spleen, stomach, and the
small and large intestine were carefully removed, and the abdominal wall was
closed. After 30 min. 40 uCi IAP were injected i.v. Twenty-ul arterial blood
samples were drawn before and every 15 min following IAP. Ninety min after
injection, the last blood samples were drawn, and the tumors were resected and
deep frozen. iaC concentrations in blood and tumor tissue were determined as
described above. The tissue-blood partition coefficient of IAP was calculated
as:
A= Ci(T)/Ca(T)
Experimental Protocol. In hamsters bearing two A-Mel-3 tumors, one of
the tumors was randomly choosen for treatment with HESW; the other served
as an imraindividual. untreated control. After exposure to HESW the animals
were randomly assigned to four groups. Tumor blood tlow was measured 30
min after treatment with HESW in the first group (n = 5). l h after HESW in
the second (n = 5). after 3 h in the third (n = 5). and after 12 h in the fourth
group (n = 6).
Statistics. For each investigated group the median blood flow ±SE was
calculated using the mean values of each single tumor.
Blood flow values in the control tumors of the different groups or in the
different groups of HESW-treated tumors were analyzed for statistical signif
icance using the Kruskal-Wallis test for nonparametric one-way analysis of
variance and multiple comparisons on ranks for independent samples (21).
Tumors treated with HESW and their corresponding controls were statistically
compared with the Wilcoxon matched pairs signed rank test. This test was also
used to compare values measured in ROI l and 2 of the same tumors (22). The
relationship between blood flow in the control tumors and MAP was examined
by linear regression and correlation analysis (22). P < 0.05 was regarded to be
significant.
RESULTS
Tissue-Blood Partition Coefficient of Iodo['4C]antipyrine. Be
tween 60 and 90 min after injection of IAP. its concentration in blood
did not change further. It was assumed therefore that an equilibrium
had been reached in the IAP distribution between blood and tissue. As
assessed by autoradiography. the distribution of IAP within the tumors
was homogeneous.
The blood-tissue partition coefficient (A) of IAP in the tumors was
0.86 ±0.06 (mean ±SD). A = 0.86 was later used for the determi
nation of tumor blood flow in control and HESW-treated tumors.
MAP during Blood Flow Measurements. MAP values in the
different groups during the injection of IAP are shown in Table I in
detail. At the beginning of IAP injection the MAP was 92.5 ±4.9 mm
Hg (median ±SE of all animals). MAP and blood flow of the control
tumors correlated significantly. Measurements reflected no signifi
cant differences in MAP between the experimental groups. MAP
remained unchanged during the injection of IAP and withdrawal of
blood samples.
1591
TI MOR PERFUSION AFTER SHOCK WAVES
Table I Middle arterial Mood pressure during injection of IAP Imm Hgl
MAP as measured through a catheter in the femoral artery at the beginning (/ = 0 si,
during U = 15 s), and the end (/ = 30 s) of IAP injection and release of arterial blood
samples. The values are given as medians ±SE in mm Hg. No significant differences
between the experimental groups or between the values for different times during IAP
injection were measured.
Experimental
groups30
min after HESW in = 5)
1 h after HESW («= 5)
3 h after HESW (n = 5)
51)2Ahll after HESW In =
together (n = 20)Time
injection/ during IAP
Os=80
±7.8
97 ±17.3
94±11.5
129.7592Â.±5
±4.9=
s84 15
±6.6
92 ±16.2
93 ±8.7
149.148Â8.±5
±2.9/
30=s80
±7.2
90 ±16.2
94 ±6.9
169.158Â7.±5
±4.0
Table 2 Blood flow in control tumors
Blood flow (ml/100 g/min) in control tumors as measured in ROÕI (whole tumor) and
ROI 2 (tumor without necrolic areas). Values are medians ±SE.
Time after
application of
HESW30
5)1min (n =
h (n = 5)
3 h (n = 5)
61)2Ahll (n =
control tumors (n = 21)Blood
flow
g/min)R(OmÃl/•100
I19.0
±15.3
23.4 ±8.9
60.0 ±18.7
1245.252Â3.±4
±7.9ROI
226.7
±22.4
32.4 ±10.6
79.4 ±19.3
1327.683Â2.±5
±5.6
Blood Flow in the Control Tumors. Blood flow of the untreated
control tumors is summarized in Table 2. No statistically significant
differences were found between the groups of control tumors. Taking
all control tumors together (Fig. 2), 23.4 ±7.9 ml/100 g/min (median
±SE) were measured in ROÕ1, corresponding to the whole tumor,
and 32.5 ±5.6 ml/100 g/min in ROI 2, corresponding to the tumor
without necrotic areas. Blood flow in the control tumors and MAP
correlated significantly (P < 0.05; Spearman's correlation coefficient
= 0.57). Values measured in ROI l were significantly less than those
measured in ROI 2 (P < 0.001). Blood flow as measured in ROI l
ranged between 2 and 80 ml/100 g/min in the different tumors. Within
one tumor, maximum and minimum values ranged between 0 and 110
ml/100 g/min if measured in small ROIs including about 1 mm2 of a
tumor cross-section.
Blood Flow in the HESW-treated Tumors. We observed that the
tumors became hemorrhagic and edematous even during the applica
tion of HESW. Tumor and tumor overlaying skin maintained their
macroscopic structure and were not ulcerated after the application of
HESW. The following results are given as median ±SE (Table 3).
Treatment with HESW induced a breakdown of tumor perfusion. As
measured in ROI 1 (Fig. 3«),tumor blood flow was reduced to 1.7 ±
0.7 ml/100 g/min 30 min after application of HESW and to 1.1 ±0.9
ml/100 g/min 1 h after HESW. In ROI 1 tumor blood flow 3 h after
HESW was slightly increased to 4.1 ±1.4 ml/100 g/min. Twelve h
after exposure to HESW tumor perfusion was 11.5 ±6.9 ml/100
g/min and thus significantly higher as compared to 30 min, l h, and
3 h after treatment (P < 0.01).
The following perfusion values were measured in ROI 2 (Fig. 3b).
Thirty min and l h after exposure to HESW tumor blood flow was 2.7
±1.2 and 2.0 ±1.5 ml/100 g/min, respectively. A significant increase
of tumor perfusion (P <0.05) was assessed 3 h after HESW: 4.0 ±1.6
ml/100 g/min. Twelve h after treatment tumor perfusion further in
creased to 24.9 ±12.8 ml/100 g/min (P < 0.01 versus 30 min and 1
h after HESW). Some tumors had blood flow values exceeding those
measured in the corresponding control tumors.
Measurements of tumor perfusion in ROI l after treatment with
HESW were always significantly lower than perfusion in the corre
sponding controls (P < 0.05). In ROI 2 blood flow in the tumors
exposed to HESW was significantly reduced at 30 min, l h, and 3 h
after treatment as well (P < 0.05), whereas in the group 12 h after
HESW no significant differences of perfusion were measured between
untreated and treated tumors.
DISCUSSION
The objective of this study was to quantify changes of tumor
perfusion during the first hours after a single treatment with HESW.
We chose the amelanotic hamster melanoma A-Mel-3 (16) for our
experiments because previous studies performed on this tumor model
in our laboratory had addressed the effects of HESW on tumor mi
crocirculation (12) and tumor growth (5). A-Mel-3 is a rapidly grow
ing and well-vascularized tumor (16, 23). Vascularization occurs be
tween 4 and 10 days after implantation, with necrotic areas appearing
on the fourth day.
The experimental Dornier lithotripter XL1 used here is similar to
other commercially available Dornier models (like the MPL 9000 or
HM3) for disintegration of kidney stones or gallstones in patients.
Maximal shock wave pressures of the XL1 are higher than those of the
MPL 9000 or HM3 (factor 1.25 or 2.6) (17), but for each stone
disintegration up to 10 times more HESW than was used in our
experiments are currently applied (24).
The major advantage of the autoradiographic tissue equilibration
technique to measure blood flow is its high spatial resolution, which
100
00
oo
o
•¿u
oom
80
60
40
20
O
I
t
JROI
1 ROI 2
Fig. 2. Blood tlow values (ml/HX) g/min) of all control tumors together as measured in
ROI l (•;corresponding to the whole tumor) or 2 (V; corresponding to the whole tumor
without necrotic areas). Each symbol <•.V) represents mean blood flow in one tumor.
Horizontal lines, median values.
Table 3 Blood flow in tumors after application of HESW
Blood tlow (ml/100 g/min) in tumors at different times after application of HESW as
measured in ROI I (whole tumor) and 2 (tumor without necrotic areas). Values are
medians ±SE.
Time after HESW
treatment30
min (n = 5)
1 h (n = 5)
3 hi« = 5)
12 h (n = 6)Blood
tlow
g/min)ROI(ml/100
l1.7
±0.7
Â1.±10.94.1
±1.4
11.5 ±6.9'ROI
22.7
±1.2"
2.0 ±1.5"
4.0 ±1.6"'*
24.9 ±12.8d
" P < 0.05 versus corresponding control tumors.
* P < 0.05 versus tumors 30 min or l h after treatment.
' P < 0.01 versus tumors 30 min. l h, or 3 h after treatment.
'' P < 0.01 versus tumors 30 min or I h after treatment.
1592
TUMOR PERFUSION AI-TER SHOCK WAVES
gg
00
oo
100
80
2 so
—¿ 40
ì
C
"O
oom
20
0
100
#
80
G
l
00
o
° 60
40
O
O
3
20
0
30 min Ih 3h 12 h
B
30 min 1h 3h 12 h
Fig. 3. Blood flow values (ml/100 g/min) of tumors treated with HESW 30 min. l h.
3 h, and 12 h after treatment, a. measurements in ROI l (corresponding to the whole
tumor). Each symbol (•) represents mean blood flow in one tumor. Horizontal lines.
median values of each group. #, P < 0.01 versus tumors 30 min. l h. and 3 h after
treatment, b, measurements in ROI 2 (corresponding to the tumor without necrotie areas).
Each symbol (T) represents mean blood flow in one tumor. Horizontal lines, median
values of each group. #.Pversus tumors 30 min or l h after treatment.
allows blood now determination in demarcated tissue volumes (25).
Horton et al. (26) compared this method with the microspheres tech
nique and found that both provide comparable perfusion values in the
brain. This autoradiographic technique has been applied to measuring
blood flow in some brain tumors (27-29) and RT-9 tumors implanted
s.c. (30). Recently, Tozer and Morris (31) measured blood flow in
LBDS fibrosarcomas implanted s.c. and different tissues of rats with
1AP autoradiography and stated that the technique provides "reason
able values" for tumor and normal tissues.
For assessing blood flow in A-mel-3 tumors we took into account
the recommendations of Patlak (32) and Williams et al. (33) for
accurate measurements: use of a short, freely flowing catheter for the
withdrawal of arterial blood, an experimental time of T = 30 s, and
fast removal and freezing of the tissue samples. Theoretically, errors
in the estimation of blood flow with IAP have to be expected in tissues
under ischemie conditions. Potential error sources are changes in the
factors A (i.e., the blood-tissue partition coefficient of IAP) and m
(i.e., the extent to which IAP diffusional equilibrium is established
between tissue and blood) because of modifications in tissue compo
sition or changes in vascular permeability, respectively (34). Marked
differences in Awould also lead to a heterogeneous distribution of IAP
between perfused and ischemie regions in untreated tumors after 90
min of equilibration time. Since we did not detect any regional het
erogeneities in IAP concentration between necrotie and vital regions
of untreated tumors in the experiments for determination of A, we
exclude major changes of A in ischemie tissues. Inaccuracy in the
determination of A(10-15%) would result in small, tolerable errors in
the calculation of blood flow (27, 32, 35). Changes in m during
ischemia are more difficult to assess. An increase in the permeability
of vessels would not affect the results since this would just shift m
toward unity, whereas m = 1 had been already assumed. If there is an
incomplete mixing of the tracer along vessels with low flow condi
tions, reduction of the true m would lead to underestimation of blood
flow (34). To date we are not aware of any changes of m during
ischemia.
The blood flow values we measured in control tumors were com
parable to those reported for other experimental tumors implanted s.c.
(30, 36. 37). Mean blood flow values of the control tumors were found
to vary within a wide range. Perfusion was also regionally heteroge
neous within each tumor. Both findings are characteristic for tumor
blood flow (36, 38). By intravital microscopy. Endlich et al. (23)
determined the following total perfusion values for A-Mel-3 tumors:
40.4 and 21.1 ml/100 g/min on the 4th and 12th days after tumor
implantation, respectively. These data correspond to our measure
ments. In control tumors as well as in tumors exposed to HESW the
blood flow values assessed in ROI l (whole tumor) were significantly
different from those of ROI 2 (tumor without necrotie regions) since
necrotie areas were in general characterized by low perfusion values.
Such relations between histology and blood flow have been described
by Tozer and Morris (31 ), Kuhnle et al. (25), and Walenta et al. (39).
Measurements in ROI 2 reflect perfusion of the vital tumor regions.
In this study, HESW had been focused on one tumor at a distance
of 3 cm from the intraindividual control tumor in the same animal. The
possibility cannot be completely excluded, however, that the control
tumor and/or the tissue surrounding the control tumor were affected
by HESW. The assumption that HESW had no relevant effect on the
perfusion of the control tumors is supported by the facts that their
blood flow values were in the range as expected for tumors implanted
s.c. and that no significant differences in perfusion rates were mea
sured that depended on the time after application of HESW.
Thirty min and l h after application of HESW tumor blood flow
was significantly reduced to values which were not clearly discernible
from the background level. Ischemia was induced in the whole tumor,
and maximum blood flow values within one tumor did not exceed 7
ml/100 g/min as measured in small ROIs (1 mm2; data not shown in
detail). Reduction of tissue perfusion is a consequence of HESWinduced
damage of tumor microcirculation. Some of the effects of
HESW on renal and other tissues are hemorrhage, edema, venous
thrombosis, and focal necrosis (7, 40). Histological and electron mi
croscopic studies have revealed defects and loss of endothelial cells,
rips in capillaries and venular walls with extravasation of red blood
cells and leukocytes, and formation of platelet plugs (10). These
morphological changes of renal vasculature cause the reduction of
renal plasma flow (8). which may become permanent (24, 41). By
intravital microscopy of the microcirculation of the dorsal skin of
hamsters, arteriolar vasoconstriction, venular hemorrhages, and
thrombus formation have been documented following exposure to
HESW (9). Similar damaging effects of HESW on tumor microcircu
lation have to be expected. Indeed, interstitial hemorrhage and vessel
damage in tumors after the application of HESW have been described
(5, 11, 12,42).
1593
TUMOR PERFUSION AFTER SHOCK WAVES
Three h after treatment a significant increase of blood flow was
measured in ROI 2 as compared to the values 30 min or l h after
HESW. This finding might be interpreted as a beginning of tumor
reperfusion. It should be noted, however, that measurements in ROI 2
might overestimate blood flow after HESW. This is due to the fact that
we were not able to determine whether the necrotic areas, which are
excluded in ROI 2, had increased in size as a consequence of HESW
or not. Thus ROI l might be more reliable for the analysis of tumor
perfusion in the treated tumors. As measured in both ROI l and 2,
tumor blood flow 12 h after the application of HESW was signifi
cantly higher as compared to earlier measurements. Values obtained in
ROI 2 indicate no differences in perfusion rates of treated and control
tumors beyond 12 h after exposure to HESW. leading to the conclu
sion that tumor reperfusion had started between 3 and 12 h after
treatment. Possible explanations for the reperfusion of the A-Mel-3
tumors are the relaxation of long-lasting vasoconstriction in the sup
plying arterioles or recanalization of thrombosed vessels. Changes in
tumor blood flow after the application of HESW were not dependent
upon macrohemodynamic parameters as suggested by MAP, which
was the same in all groups. To exclude the influence of systemic
effects on blood flow measurements, tumors after the application of
HESW were compared to intraindividual control tumors; the break
down of tumor perfusion and early reperfusion between 3 and 12 h
after HESW were confirmed.
Blood flow reduction after HESW is probably one of the main
mechanisms leading to the delay of tumor growth. The findings of
Oosterhof et al. (6) that HESW are more effective on well vascularized
tumors support this statement. Tumor cell death secondary to
ischemia plays an important role in therapeutic modalities like hyperthermia
(13, 15, 36) or photodynamic therapy (14. 15) and could be
essential for tumor therapy with HESW. In addition to focusing the
shock waves on the tumor, an increased sensitivity of tumor vasculature
to the treatment could constitute a factor enhancing its selective
action.
We postulate that repeated applications of HESW in short intervals,
i.e., before tumor reperfusion after each exposure occurs, would pro
long tumor ischemia and have a more pronounced therapeutic effect.
Indeed, the same number of HESW is more effective if applied in
many fractionated doses, as shown by Oosterhof et al. (6) and Hoshi
et al. (42) or Weiss et al. (5) for A-Mel-3 tumors. However, complete
tumor remission after repeated applications of HESW has not been
achieved yet. Since the extent of perfusion defects and the time
needed for reperfusion had not been considered in those studies we
suppose that the intervals chosen between the exposures to HESW
(12, 24, or 48 h) had been too long.
The effects of HESW on tumor blood flow must also be taken into
account for combined treatment with other agents like chemotherapeutics.
Several studies have demonstrated additive and/or synergistic
effects of HESW and chemotherapeutic agents or biological response
modifiers (43-45). According to our results, the chemotherapeutic
agent must be given prior to the application of HESW to make
possible its intravascular transport into the tumor. On the other hand,
if HESW are applied after the chemotherapeutic agent has accumu
lated in the tumor the blood flow reduction induced would contribute
to a slower washout of the agent. Based on our knowledge, only
agents that are active under ischemie conditions should be considered.
We conclude that HESW have significant effects on tumor perfu
sion which most probably determine their therapeutic efficiency. Per
fusion changes should be taken into account to optimize tumor therapy
with HESW and/or the combined treatment with HESW and other
therapeutic strategies.
ACKNOWLEDGMENTS
The authors gratefully acknowledge the valuable comments of Prof. Dr. K.
Messmer, Prof. Dr. W. Mueller-Klieser, and S. Walenta on the manuscript.
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1989.
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tumors in vivo. Ultrasound Med. Biol., 16: 595-605. 1990.
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In vivo effects of high energy shock waves on urological tumors: an evaluation of
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Prog. Appi. Microcirc.. 12: 41-50. 1987.
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Vick, N. A. Regional measurements of blood flow in experimental RG-2 rat gliomas.
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Pixel-to-pixel correlation between images of absolute ATP concentrations and blood
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sound Med. Biol., 14: 117-122, 1988.
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Extracorporeal shock-wave lithotripsy: long-term complications. Am. J. Roentgenol.,
ISO: 311-315. 1988.
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Cancer Res.. 83: 248-250. 1992.
1595

frog

1. Introduction
In the mythological age, it was told that the frog, after having perfected plans to employ
the services of its offsprings, went to challenge the hare for a 12km race. He had boasted
that it would win the race at the hare’s expense, after having ambushed each offspring at
every kilometre end of course. The hare, ignorant of the intrigues, counted the challenge
as an insult but accepted all the same to partake in the race. It was at the finish-point
that he realised that he had been fooled. I read this ”frog-hare” mythology as a young
man and had ever since thought that any life race could be so won. The fun of the frog’s
victory, in part, motivated me in writing this paper, dedicated to its random movement
in a quadrangle.
The life personae of frog and hare can be encapsulated in the following. The frog is
an amphibian small animal with long back legs for jumping. It is tailess, no wonder then,
”it is God that provides protection for its fly menace”, so goes the dictum. The hare is
an animal like a large rabbit with very strong back legs, known for runnning fast. From
the foregoing, it could be concluded that the frog cannot in any form be a match for the
hare in any race whatsoever, except possibly, in the water.
The other leg of motivation for this work is that the random jumps constitute an aspect
of random motion/fractal movement. Theory of random walk, fractal motion or brownian
movement have close relationship with mathematical theory of Graphs which has wide applications
in electrical and civil engineering, communication networks, industrial management,
operations research, computer science, economics, management science, marketing,
sociology and others.
Infact, concept of graphs can be encountered under variety of names such as network
1
in electrical/electronics engineering, structures in civil engineering, molecular structures
in chemistry, organisational structures in economics or management science, sociograms,
road maps in transportation, etc. Some of the specific problems being examined include
minimum cost or time in transportation, best assignment of workers to jobs, most efficient
use of telephone networks, shortest path to a destination. Essentially, most of these lead
to optimisation processes. Hence, there is the need to obtain essential statistical parameters
to authenticate correctness or efficiency of models. This work examines the random
jumps of a frog in a quadrangle and introduces a method for obtaining the associated statistical
parameters such as expectation, mean deviation, moment, skewness and kurtosis.
The obtained parameters are compared with those obtained by other standard methods.
2. Statement of problem
A frog displaces randomly by following the lignes of a a quadrangle with nine points as
indicated in the figure below. At each of these nine points, it randomly chooses a direction
of movement. Hence, from point O, the four points OAi has each a probability of 1/4 of
being followed by the frog. From point Ai, the three directions has each a probability of
1/3 to be followed, and from point Bi, each of the two directions has a probability of 1/2.
The passage from one point to another in the quadrangle is called ”a path” of the frog.
We assume that the frog starts from 0. We are interested in the probability that the frog
returns to 0 for the first time after K movements, where K is a positive integer.
3. Methodology:
Path graph of the problem:
2
Now, we set out to propose (expose) the mathematical theory to obtain the vital statistical
parameters (Moment generating function, skewness and kurtosis) for the jumps
We consider the probability of each path of the frog as follows, using arrows to indicate
directions of movement in the quadrangle.
Simplification of path graph:
We construct a simpler (representative) path for the frog. For this, we consider the frog’s
movement to and fro each of the following:
3
(a) central point O;
(b) a lateral point A;
(c) a corner B
The probability graph of these points is represented below as:
where
(i) probability of out-movement from O is 1;
4
,
(ii) probability of out-movement from A to B is 2/3;
(iii) probability of out-movement from B is 1;
(iv) probability of out-movement from A to O is 1/3.
Probability of jump:
Now, define: P(k), the probability that the frog returns to point O after k steps since
it departed from it.
Clearly, P[OA] = P[AB] = 1.
P[AO] = 1/3
P[AB] = 2/3
Theorem 1:
Arrival of the frog to point O (after its initial departure from it) takes place after even
number of movements i.e K = 2n , n 2 N ,and P(k) = 1/3(2/3)n−1.
5
Proof:
For the frog to jump from O to A and return to O, for example, it needs to jump OA and
back-jump AO: K = 2.
For the frog to want to return from B to O, it must have made the trajectories: OA, AB,
BA and AO: K = 4.
Generally, to return to O (from B), it must undergo the trajectory OA, next (n − 1)
go-come of AB, BA and lastly, AO: K = 2n.
P[K] = P[2n] = P(OA) × P(AB) × P(BA) × P(AO)
= 1 × (2/3)n−1 × 1n−1 × 1/3
= 1/3 × (2/3)n−1
Verification:
1X
n=1
P(k) =
1X
n=1
1
3
(
2
3
)n−1
=
1
3
+ (
2
3
) + . . .
=
1
3
1 − 2
3
= 1
Introduction of differential calculus technique. Before continuing this study, it is
here appropriate to expose and explain a mathematical approach used to obtain the parameters
of the frog’s random jumps.
Now, consider the probaility P(k) as a sequence of probability function f(x) where x is a
random k variable. Then we invoke the theorem of calculus on sum of function on series
of {fk(x)}:
Theorem M: The derivatives of a sum of functions is equal to the sum of derivatives of
6
functions: i.e.
d
dx

X
k
fk(x)
!
=
X
k
d
dx
f(x)
Proof: From l.h.s.
X
k
fk(x) = f1(x) + f2(x) + · · · + fk(x) + · · · (1)
Let the operator d
dx (.) be used on (1), we have:
d
dx

X
k
fk(x)
!
=
d
dx
f1(x) +
d
dx
f2(x) + · · · +
d
dx
fk(x) + · · · (2)
=
X
k
d
dx
f(x) · · · (3)
(3) constitutes the r.h.s. of the theorem
Remark: Theorem M can be extended to cover the case dh
dxh (
P
k fk(x)), where dh
dxh (•) is the
hth derivative of the function fk(x). In this case, we have
dh
dxh

X
k
fk(x)
!
=
X
k
dh
dxh (fk(x))
Statistical parameters of jumps.
Moment Generating Function of X:
Theorem 2: Define X: number of jumps required of the frog to have been made to return
to point O, after having departed from it.
Then E(X) = 6, X 2 {2, 4, . . . , 2n, . . .}
Proof:
E(X) =
1X
n=1
2n
1
3
(
2
3
)n−1
=
2
3
1X
n=1
n(
2
3
)n−1
7
let x = 2/3
E(X) =
2
3
1X
n=1
d
dx
(xn)
=
2
3
d
dx
1X
n=1
xn
!
=
2
3
d
dx

x
1 − x

=
2
3
"
(1 − x) + x
(1 − x)2
#
=
2
3
"
1
(1 − x)2
#
=
2
3
×
1
(1/3)2
=
2
3
×
9
1
= 6
Definition: Since X is a discrete random variate, moment generating function of X is
defined for any real number h:
M(h) = E

ehx

(a)
=
1X
n=1
e2nh 1
3
(
2
3
)n−1
=
1X
n=1

1 + 2nh +
(2nh)2
2!
+ . . .
!
1
3
(2/3)n−1
=
1X
n=1
1
3
(2/3)n−1 +
1X
n=1
(2n)r hr
r!
1
3
(2/3)n−1
= 1 +
1X
n=1
(2n)r hr
r!
1
3
(2/3)n−1
= 1 +
1X
n=1
(X)r hr
r!
P (X = 2n) (b)
Definition: Define μ0
r = E (Xr) =
P
(2n)r 1
3 (2/3)n−1, the rth moment of X about the
origin.
8
μ0
r: first moment about the origin, i.e μ0
r = E (X) = μ, the expectation of X obtained in
theorem 2.
4. New Approach for Evaluating Statistical Parameters of Jump:
Hitherto, evaluation of the moment generating functions of random variates is carried
out using the infinite geometric approach. But here, we propose an alternative approach
which adopts the linearity of summation 0P0 property and the differentiation technique.
We subsequently elucidate on this. In fact, using the ”new” approach, we define
μ
0
r = E

X2

=
1X
n=1
4n2 1
3
(2/3)n−1
=
4
3
X
n (n − 1 + 1) (2/3)n−1
=
4
3
X
n (n − 1) (2/3)n−1 +
4
3
X
n (2/3)n−1
=
4
3
d2
dx2 (xn) +
4
3
× 9
=
4
3

2
(1 − x)3
!
+ 12
=
4
3
×
2
1
×
27
1
+ 12
= 72 + 12 = 84.
Hence, the population variance of the frog’s number of steps required of it to return to
O,for the first time after departure is:
2 = μ2 − μ2 = 84 − 36 = 48
Now, the : population standard deviation of the frog’s number of return jumps to the
origin
 =
p
48 = 4
p
3.
9
Skewness of Jump
It is desired to study whether the distribution of these jumps is symmetrical or skewed
about its mean. That is, it is of interest to veriff whetehr or not the natural assumption
that the frog makes random (unconstrained) movement at each point could lead to an
absolutely normal (i.e. bell-shaped) distribution, without skewness of any sort.
Now, define
μr = E[(x − μ)r] =
X
(2n − 6)r 1
3
(
2
3
)n−1 (c)
This is called the rth moment of the variate X about the mean.
Now, for a symmentrical distribution, μ3 = 0. Also,
skewness =
μ3
3 ,
where
μ3
3 > 0
means a skew to the right in which case, the distribution tapers off to the right more than
to the left, and vice-versa, otherwise. Now,
μ3 = μ
0
3 − 3μ
0
2μ + 2μ3,
obtained by expanding equation (c). Now, using (b),
μ
0
3 =
8
3
X
n3(
2
3
)n−1.
Using the identity
n3 = (n + 1)n(n − 1) + n,
10
μ
0
3 =
8
3
X
(n + 1)n(n − 1)(
2
3
)n−1 +
8
3
X
n(
2
3
)n−1
μ
0
3 =
16
9
d3
dx3 (xn+1) +
8
3
d
dx
(xn) = 888
for x = 2
3 . Hence,
μ3 = 888 − 3(84)(6) + 2(6)3 = 192.
Hence, the distribution of X is skewed to the right, as dividing μ3 by 3 does not change
its positivity.The division merely divests the distribution of its units of measurement.
Hence, the skewness = μ3
3 is a pure number. Thus
192
(4
p
3)3
=
192
332.55
= 0.577
is the skewness factor of the distribution.
Kurtosis It is of much interest also to calculate the kurtosis of the distribution of X,
the number of steps needed by the frog to have been realized before finding itself back at
11
the origin, after initial departure. Again the natural assumption of normal distribution,
encoded in randomness assumption of the jumps at every point, presuposes normalized
kurtosis. Thus, it is of valuable information to ascertain if this is so or not.
The fourth moment, μ4, divided by μ22
is a measure of kurtosis, the peakedness or ”pinpointedness”
of the central peak of the curve of the distribution. It is generally observed
that the bigger the μ4 is, the higher the central peak of the distribution becomes.
Now,
μ
0
4 =
X
(2n)4 1
3
(
2
3
)n−1 =
16
4
X
n4(
2
3
)n−1
From the identity
n4 = (n + 2)(n + 1)n(n − 1) − 2(n + 1)n(n − 1) + n2
we have
μ
0
4 =
16
3
X
(n+2)(n+1)n(n−1)(
2
3
)n−1 −
32
3
X
(n+1)n(n−1)(
2
3
)n−1 +
16
3
X
n2(
2
3
)n−1
=
32
9
d4
dx4 (xn+2) −
64
9
d3
dx4 (xn+1) + 4μ
0
2 = 2064
μ4 = μ
0
4 − 4μ
0
3μ + 6μ
0
2μ2 − 3μ4
= 2064 − 4  888  6 + 6  84  36 − 3  362
= 2064 − 21312 + 18144 − 3888
= −4992
Hence,
μ4
μ22
=
−4992
482 = −2.167
12
a negative central peak.
Results:
The following results have been obtained from this study:
(1) It is revealed that the distribution of the frog’s first return to the ’origin’ is not
symmetrical about its mean;
(2) It is also found out that the kurtosis of the movement is negative;
(3) This study paves way for expressing nk (k, a positive integer) as sum of factorial
products of n as exemplified by
1. n1 = n
2. n2 = n(n − 1) + n
3. n3 = (n + 1)n(n − 1) + n
4. n4 = (n + 2)(n + 1)n(n − 1) − 2(n + 1)n(n − 1) + n2
= (n + 2)(n + 1)n(n − 1) − 2(n + 1)n(n − 1) + n(n − 1) + n
5. n5 = (n+3)(n+2)(n+1)n(n−1)−5(n+2)(n+1)n(n−1)+5(n+1)n(n−1)+n
...
...
...
...
... ...
...
nk = ?
Conclusion
That the distribution of the frog’s first return to the ”origin” after its initial departure
is not symmetric, is strange. A zero kurtosis would probably have meant that the distribution
is ”peakedly normal”. But here, we have a negative kurtosis. Effort would be
13
made in future studies to sort for an explanation on how and why all these ”strange”
observations come about.
Literature
(1) T.J.Adesakin, A.A.Osuntunyi and M.A.Olagunju: The distributional properties of the
family of logistic Distributions: Ife Journal of science. Special Edition 2008, Vol.10 no 1
pgs 245−
(2) Introduction to statistical inference E.S.Keeping.(university of Alberta)
(3) Outils Mathematiques probabilities: Bernard Vauquois(Hermann collection Paris methods)
(4) Aide memoire de mathematiques superieures M.Vygodski(Edition de Moscov).
14

Techniques of Integration

Integration by formulae
There exist many books that contain extensive lists of integration, differentiation
and other mathematical formulae. For our purpose we will use the
list given below.
1.
Z
af(u)du = a
Z
f(u)du
2.
Z
Xn
i=1
aifi(u)
!
du =
Xn
i=1
Z
aifi(u)du

3.
Z
undu =
un+1
n + 1
+ C, n 6= −1
4.
Z
u−1du = ln |u| + C
5.
Z
eaudu =
e6au
a
+ C
6.
Z
abudu =
abu
b ln a
+ C, a > 0, a 6= 1
7.
Z
ln |u|du = u ln |u| − u + C
267
268 CHAPTER 6. TECHNIQUES OF INTEGRATION
8.
Z
sin(au)du =
−cos(au)
a
+ C
9.
Z
cos(au)du =
sin(au)
a
+ C
10.
Z
tan(au)du =
ln | sec(au)|
a
+ C
11.
Z
cot(au)du =
ln | sin(au)|
a
+ C
12.
Z
sec(au)du =
ln | sec(au) + tan(au)|
a
+ C
13.
Z
csc(au)du =
ln | csc(au) − cot(au)|
a
+ C
14.
Z
sinh(au)du =
cosh(au)
a
+ C
15.
Z
cosh(au)du =
sinh(au)
a
+ C
16.
Z
tanh(au)du =
ln | cosh(au)|
a
+ C
17.
Z
coth(au)du =
ln | sinh(au)|
a
+ C
18.
Z
sech (au)du =
2
a
arctan(eau) + C
19.
Z
csch (au) du =
2
a
arctanh (eau) + C
20.
Z
sin2(au)du =
u
2
−
sin(au) cos(au)
2a
+ C
21.
Z
cos2(au)du =
u
2
+
sin(au) cos(au)
2a
+ C
22.
Z
tan2(au)du =
tan(au)
a
− u + C
6.1. INTEGRATION BY FORMULAE 269
23.
Z
cot2(au)du = −
cot(au)
a
− u + C
24.
Z
sec2(au)du =
tan(au)
a
+ C
25.
Z
csc2(au)du = −
cot(au)
a
+ C
26.
Z
sinh2(au)du = −
u
2
+
sinh(2au)
4a
+ C
27.
Z
cosh2(au)du =
u
2
+
sinh(2au)
4a
+ C
28.
Z
tanh2(au)du = u −
tanh(au)
a
+ C
29.
Z
coth2(au)du = u −
coth(au)
a
+ C
30.
Z
sech 2(au)du =
tanh(au)
a
+ C
31.
Z
csch 2(au)du =
−coth(au)
a
+ C
32.
Z
sec(au) tan(au)du =
sec(au)
a
+ C
33.
Z
csc(au) cot(au)du = −
csc(au)
a
+ C
34.
Z
sech (au) tanh(au)du = −
sech (au)
a
+ C
35.
Z
csch (au) coth(au)du = −
csch (au)
a
+ C
36.
Z
du
a2 + u2 =
1
a
arctan
u
a

+ C
37.
Z
du
a2 − u2 =
1
a
arctanh
u
a

+ C =
1
2a
ln

a + u
a − u

+ C
270 CHAPTER 6. TECHNIQUES OF INTEGRATION
38.
Z
du
p
a2 + u2
= arcsinh
u
a

+ C
39.
Z
du
p
a2 − u2
= arcsin
u
a

+ C, |a| > |u|
40.
Z
du
p
u2 − a2
= arccosh
u
a

+ C, |u| > |a|
41.
Z
du
u
p
u2 − a2
=
1
a
arcsec
u
a

+ C, |u| > |a|
42.
Z
du
u
p
a2 − u2
= −
1
a
arcsech
u
a

+ C, |a| > |u|
43.
Z
du
u
p
a2 + u2
= −
1
a
arccsch
u
a

+ C
44.
Z
u du
p
a2 + u2
=
p
a2 + u2 + C
45.
Z
u du
a2 − u2 = −ln
p
a2 − u2 + C, |a| > |u|
46.
Z
u du
p
a2 + u2
=
p
a2 + u2 + C
47.
Z
u du
p
a2 − u2
= −
p
a2 − u2 + C, |a| > |u|
48.
Z
u du
p
u2 − a2
=
p
u2 − a2 + C, |u| > |a|
49.
Z
arcsin(au)du = u arcsin(au) +
1
a
p
1 − a2u2 + C, |a||u| < 1
50.
Z
arccos(au)du = u arccos(au) −
1
a
p
1 − a2u2 + C, |a||u| < 1
51.
Z
arctan(au)du = u arctan(au) −
1
2a
ln(1 + a2u2) + C
52.
Z
arccot (au)du = uarccot (au) +
1
2a
ln(1 + a2u2) + C
6.1. INTEGRATION BY FORMULAE 271
53.
Z
arcsec (au)du = uarcsec (au) −
1
a
ln

au +
p
a2u2 − 1

+ C, au > 1
54.
Z
arccsc (au)du = uarccsc (au) +
1
a
ln

au +
p
a2u2 − 1

+ C, au > 1
55.
Z
arcsinh (au)du = uarcsinh (au) −
1
a
p
1 + a2u2 + C
56.
Z
arccosh (au)du = uarccosh (au) −
1
a
p
−1 + a2u2 + C, |a||u| > 1
57.
Z
arctanh (au)du = uarctanh (au) +
1
2a
ln(−1 + a2u2) + C, |a||u| 6= 1
58.
Z
arccoth (au)du = uarccoth (au) +
1
2a
ln(−1 + a2u2) + C, |a||u| 6= 1
59.
Z
arcsech (au)du = uarcsech (au) +
1
a
arcsin(au) + C, |a||u| < 1
60.
Z
arccsch (au)du = uarccsch (au) +
1
a
ln

au +
p
a2u2 + 1

+ C
61.
Z
eau sin(bu)du =
eau[a sin(bu) − b cos(bu)]
a2 + b2 + C
62.
Z
eau cos(bu)du =
eau[a cos(bu) + b sin(bu)]
a2 + b2 + C
63.
Z
sinn(u)du =
−1
n

sinn−1(u) cos(u)

+
n − 1
n
Z
sinn−2(u)du
64.
Z
cosn(u)du =
1
n

cosn−1(u) sin(u)

+
n − 1
n
Z
cosn−2(u)du
65.
Z
tann(u)du =
tann−1(u)
n − 1
−
Z
tann−2(u)du
66.
Z
cotn(u)du = −
cotn−1(u)
n − 1
−
Z
cotn−2(u)du
67.
Z
secn(u)du =
1
n − 1

secn−2(u) tan(u)

+
n − 2
n − 1
Z
secn−2(u)du
272 CHAPTER 6. TECHNIQUES OF INTEGRATION
68.
Z
cscn(u)du =
−1
n − 1

cscn−2(u) cot(u)

+
n − 2
n − 1
Z
cscn−2(u)du
69.
Z
sin(mu)sin(nu)du =
sin[(m − n)u]
2(m − n)
−
sin[(m + n)u]
2(m + n)
+ C, m2 6= n2
70.
Z
cos(mu) cos(nu)du =
sin[(m − n)u]
2(m − n)
+
sin[(m + n)u]
2(m + n)
+ C, m2 6= n2
71.
Z
sin(mu) cos(nu)du =
cos[(m − n)u]
2(m − n)
−
cos[(m + n)u]
2(m + n)
+ C, m2 6= n2

Sets and Functions

Sets and Functions
1.1 Sets
Denition. A set is a collection of objects. We call the objects, elements. A set is denoted by listing the elements
between braces. For example: fe; {; ; 1g. We use ellipses to indicate patterns. The set of positive integers is
f1; 2; 3; : : :g. We also denote a sets with the notation fxjconditions on xg for sets that are more easily described than
enumerated. This is read as \the set of elements x such that x satises . . . ". x 2 S is the notation for \x is an
element of the set S." To express the opposite we have x 62 S for \x is not an element of the set S."
Examples. We have notations for denoting some of the commonly encountered sets.
 ; = fg is the empty set, the set containing no elements.
 Z = f: : : ;􀀀1; 0; 1 : : :g is the set of integers. (Z is for \Zahlen", the German word for \number".)
 Q = fp=qjp; q 2 Z; q 6= 0g is the set of rational numbers. (Q is for quotient.)
 R = fxjx = a1a2


an:b1b2


g is the set of real numbers, i.e. the set of numbers with decimal expansions. 1
1Guess what R is for.
2
 C = fa + {bja; b 2 R; {2 = 􀀀1g is the set of complex numbers. { is the square root of 􀀀1. (If you haven't seen
complex numbers before, don't dismay. We'll cover them later.)
 Z+, Q+ and R+ are the sets of positive integers, rationals and reals, respectively. For example, Z+ = f1; 2; 3; : : :g.
 Z0+, Q0+ and R0+ are the sets of non-negative integers, rationals and reals, respectively. For example, Z0+ =
f0; 1; 2; : : :g.
 (a : : : b) denotes an open interval on the real axis. (a : : : b)  fxjx 2 R; a < x < bg
 We use brackets to denote the closed interval. [a : : : b]  fxjx 2 R; a  x  bg
The cardinality or order of a set S is denoted jSj. For nite sets, the cardinality is the number of elements in the
set. The Cartesian product of two sets is the set of ordered pairs:
X  Y  f(x; y)jx 2 X; y 2 Y g:
The Cartesian product of n sets is the set of ordered n-tuples:
X1  X2 


 Xn  f(x1; x2; : : : ; xn)jx1 2 X1; x2 2 X2; : : : ; xn 2 Xng:
Equality. Two sets S and T are equal if each element of S is an element of T and vice versa. This is denoted,
S = T. Inequality is S 6= T, of course. S is a subset of T, S  T, if every element of S is an element of T. S is a
proper subset of T, S  T, if S  T and S 6= T. For example: The empty set is a subset of every set, ;  S. The
rational numbers are a proper subset of the real numbers, Q  R.
Operations. The union of two sets, S [ T, is the set whose elements are in either of the two sets. The union of n
sets,
[nj
=1Sj  S1 [ S2 [


[ Sn
is the set whose elements are in any of the sets Sj . The intersection of two sets, S \ T, is the set whose elements are
in both of the two sets. In other words, the intersection of two sets in the set of elements that the two sets have in
common. The intersection of n sets,
\nj
=1Sj  S1 \ S2 \


\ Sn
3
is the set whose elements are in all of the sets Sj . If two sets have no elements in common, S \ T = ;, then the sets
are disjoint. If T  S, then the dierence between S and T, S n T, is the set of elements in S which are not in T.
S n T  fxjx 2 S; x 62 Tg
The dierence of sets is also denoted S 􀀀 T.
Properties. The following properties are easily veried from the above denitions.
 S [ ; = S, S \ ; = ;, S n ; = S, S n S = ;.
 Commutative. S [ T = T [ S, S \ T = T \ S.
 Associative. (S [ T) [ U = S [ (T [ U) = S [ T [ U, (S \ T) \ U = S \ (T \ U) = S \ T \ U.
 Distributive. S [ (T \ U) = (S [ T) \ (S [ U), S \ (T [ U) = (S \ T) [ (S \ U).
1.2 Single Valued Functions
Single-Valued Functions. A single-valued function or single-valued mapping is a mapping of the elements x 2 X
into elements y 2 Y . This is expressed as f : X ! Y or X
f!
Y . If such a function is well-dened, then for each
x 2 X there exists a unique element of y such that f(x) = y. The set X is the domain of the function, Y is the
codomain, (not to be confused with the range, which we introduce shortly). To denote the value of a function on a
particular element we can use any of the notations: f(x) = y, f : x 7! y or simply x 7! y. f is the identity map on
X if f(x) = x for all x 2 X.
Let f : X ! Y . The range or image of f is
f(X) = fyjy = f(x) for some x 2 Xg:
The range is a subset of the codomain. For each Z  Y , the inverse image of Z is dened:
f􀀀1(Z)  fx 2 Xjf(x) = z for some z 2 Zg:
4
Examples.
 Finite polynomials and the exponential function are examples of single valued functions which map real numbers
to real numbers.
 The greatest integer function, b
c, is a mapping from R to Z. bxc in the greatest integer less than or equal to x.
Likewise, the least integer function, dxe, is the least integer greater than or equal to x.
The -jectives. A function is injective if for each x1 6= x2, f(x1) 6= f(x2). In other words, for each x in the domain
there is a unique y = f(x) in the range. f is surjective if for each y in the codomain, there is an x such that y = f(x).
If a function is both injective and surjective, then it is bijective. A bijective function is also called a one-to-one mapping.
Examples.
 The exponential function y = ex is a bijective function, (one-to-one mapping), that maps R to R+. (R is the set
of real numbers; R+ is the set of positive real numbers.)
 f(x) = x2 is a bijection from R+ to R+. f is not injective from R to R+. For each positive y in the range, there
are two values of x such that y = x2.
 f(x) = sin x is not injective from R to [􀀀1::1]. For each y 2 [􀀀1; 1] there exists an innite number of values of
x such that y = sin x.
1.3 Inverses and Multi-Valued Functions
If y = f(x), then we can write x = f􀀀1(y) where f􀀀1 is the inverse of f. If y = f(x) is a one-to-one function, then
f􀀀1(y) is also a one-to-one function. In this case, x = f􀀀1(f(x)) = f(f􀀀1(x)) for values of x where both f(x) and
f􀀀1(x) are dened. For example log x, which maps R+ to R is the inverse of ex. x = elog x = log(ex) for all x 2 R+.
(Note the x 2 R+ ensures that log x is dened.)
5
Injective Surjective Bijective
Figure 1.1: Depictions of Injective, Surjective and Bijective Functions
If y = f(x) is a many-to-one function, then x = f􀀀1(y) is a one-to-many function. f􀀀1(y) is a multi-valued function.
We have x = f(f􀀀1(x)) for values of x where f􀀀1(x) is dened, however x 6= f􀀀1(f(x)). There are diagrams showing
one-to-one, many-to-one and one-to-many functions in Figure 1.2.
domain range domain range domain range
one-to-one many-to-one one-to-many
Figure 1.2: Diagrams of One-To-One, Many-To-One and One-To-Many Functions
Example 1.3.1 y = x2, a many-to-one function has the inverse x = y1=2. For each positive y, there are two values of
x such that x = y1=2. y = x2 and y = x1=2 are graphed in Figure 1.3.
6
Figure 1.3: y = x2 and y = x1=2
We say that there are two branches of y = x1=2: the positive and the negative branch. We denote the positive
branch as y =
p
x; the negative branch is y = 􀀀
p
x. We call
p
x the principal branch of x1=2. Note that
p
x is a
one-to-one function. Finally, x = (x1=2)2 since (
p
x)2 = x, but x 6= (x2)1=2 since (x2)1=2 = x. y =
p
x is graphed
in Figure 1.4.
Figure 1.4: y =
p
x
Now consider the many-to-one function y = sin x. The inverse is x = arcsin y. For each y 2 [􀀀1; 1] there are an
innite number of values x such that x = arcsin y. In Figure 1.5 is a graph of y = sin x and a graph of a few branches
of y = arcsin x.
Example 1.3.2 arcsin x has an innite number of branches. We will denote the principal branch by Arcsin x which
maps [􀀀1; 1] to

􀀀
2 ; 
2

. Note that x = sin(arcsin x), but x 6= arcsin(sin x). y = Arcsin x in Figure 1.6.
7
Figure 1.5: y = sin x and y = arcsin x
Figure 1.6: y = Arcsin x
Example 1.3.3 Consider 11=3. Since x3 is a one-to-one function, x1=3 is a single-valued function. (See Figure 1.7.)
11=3 = 1.
Figure 1.7: y = x3 and y = x1=3
8
Example 1.3.4 Consider arccos(1=2). cos x and a few branches of arccos x are graphed in Figure 1.8. cos x = 1=2
Figure 1.8: y = cos x and y = arccos x
has the two solutions x = =3 in the range x 2 [􀀀; ]. Since cos(x + ) = 􀀀cos x,
arccos(1=2) = f=3 + ng:
1.4 Transforming Equations
We must take care in applying functions to equations. It is always safe to apply a one-to-one function to an equation,
(provided it is dened for that domain). For example, we can apply y = x3 or y = ex to the equation x = 1. The
equations x3 = 1 and ex = e have the unique solution x = 1.
If we apply a many-to-one function to an equation, we may introduce spurious solutions. Applying y = x2 and
y = sin x to the equation x = 
2 results in x2 = 2
4 and sin x = 1. The former equation has the two solutions x = 
2 ;
the latter has the innite number of solutions x = 
2 + 2n, n 2 Z.
We do not generally apply a one-to-many function to both sides of an equation as this rarely is useful. Consider the
equation
sin2 x = 1:
9
Applying the function f(x) = x1=2 to the equation would not get us anywhere
(sin2 x)1=2 = 11=2:
Since (sin2 x)1=2 6= sin x, we cannot simplify the left side of the equation. Instead we could use the denition of
f(x) = x1=2 as the inverse of the x2 function to obtain
sin x = 11=2 = 1:
Then we could use the denition of arcsin as the inverse of sin to get
x = arcsin(1):
x = arcsin(1) has the solutions x = =2 + 2n and x = arcsin(􀀀1) has the solutions x = 􀀀=2 + 2n. Thus
x =

2
+ n; n 2 Z:
Note that we cannot just apply arcsin to both sides of the equation as arcsin(sin x) 6= x.
10
1.5 Exercises
Exercise 1.1
The area of a circle is directly proportional to the square of its diameter. What is the constant of proportionality?
Hint, Solution
Exercise 1.2
Consider the equation
x + 1
y 􀀀 2
=
x2 􀀀 1
y2 􀀀 4
:
1. Why might one think that this is the equation of a line?
2. Graph the solutions of the equation to demonstrate that it is not the equation of a line.
Hint, Solution
Exercise 1.3
Consider the function of a real variable,
f(x) =
1
x2 + 2
:
What is the domain and range of the function?
Hint, Solution
Exercise 1.4
The temperature measured in degrees Celsius 2 is linearly related to the temperature measured in degrees Fahrenheit 3.
Water freezes at 0 C = 32 F and boils at 100 C = 212 F. Write the temperature in degrees Celsius as a function
of degrees Fahrenheit.
2 Originally, it was called degrees Centigrade. centi because there are 100 degrees between the two calibration points. It is now
called degrees Celsius in honor of the inventor.
3 The Fahrenheit scale, named for Daniel Fahrenheit, was originally calibrated with the freezing point of salt-saturated water to
be 0. Later, the calibration points became the freezing point of water, 32, and body temperature, 96. With this method, there are
64 divisions between the calibration points. Finally, the upper calibration point was changed to the boiling point of water at 212.
This gave 180 divisions, (the number of degrees in a half circle), between the two calibration points.
11
Hint, Solution
Exercise 1.5
Consider the function graphed in Figure 1.9. Sketch graphs of f(􀀀x), f(x + 3), f(3 􀀀 x) + 2, and f􀀀1(x). You may
use the blank grids in Figure 1.10.
Figure 1.9: Graph of the function.
Hint, Solution
Exercise 1.6
A culture of bacteria grows at the rate of 10% per minute. At 6:00 pm there are 1 billion bacteria. How many bacteria
are there at 7:00 pm? How many were there at 3:00 pm?
Hint, Solution
Exercise 1.7
The graph in Figure 1.11 shows an even function f(x) = p(x)=q(x) where p(x) and q(x) are rational quadratic
polynomials. Give possible formulas for p(x) and q(x).
Hint, Solution
12
Figure 1.10: Blank grids.
Exercise 1.8
Find a polynomial of degree 100 which is zero only at x = 􀀀2; 1;  and is non-negative.
Hint, Solution
Exercise 1.9
Hint, Solution
13
1 2
1
2
2 4 6 8 10
1
2
Figure 1.11: Plots of f(x) = p(x)=q(x).
Exercise 1.10
Hint, Solution
Exercise 1.11
Hint, Solution
Exercise 1.12
Hint, Solution
Exercise 1.13
Hint, Solution
Exercise 1.14
Hint, Solution
Exercise 1.15
Hint, Solution
Exercise 1.16
Hint, Solution
14
1.6 Hints
Hint 1.1
area = constant  diameter2.
Hint 1.2
A pair (x; y) is a solution of the equation if it make the equation an identity.
Hint 1.3
The domain is the subset of R on which the function is dened.
Hint 1.4
Find the slope and x-intercept of the line.
Hint 1.5
The inverse of the function is the re
ection of the function across the line y = x.
Hint 1.6
The formula for geometric growth/decay is x(t) = x0rt, where r is the rate.
Hint 1.7
Note that p(x) and q(x) appear as a ratio, they are determined only up to a multiplicative constant. We may take the
leading coecient of q(x) to be unity.
f(x) =
p(x)
q(x)
=
ax2 + bx + c
x2 + x + 
Use the properties of the function to solve for the unknown parameters.
Hint 1.8
Write the polynomial in factored form.
15
1.7 Solutions
Solution 1.1
area =   radius2
area =

4
 diameter2
The constant of proportionality is 
4 .
Solution 1.2
1. If we multiply the equation by y2 􀀀 4 and divide by x + 1, we obtain the equation of a line.
y + 2 = x 􀀀 1
2. We factor the quadratics on the right side of the equation.
x + 1
y 􀀀 2
=
(x + 1)(x 􀀀 1)
(y 􀀀 2)(y + 2)
:
We note that one or both sides of the equation are undened at y = 2 because of division by zero. There are
no solutions for these two values of y and we assume from this point that y 6= 2. We multiply by (y􀀀2)(y+2).
(x + 1)(y + 2) = (x + 1)(x 􀀀 1)
For x = 􀀀1, the equation becomes the identity 0 = 0. Now we consider x 6= 􀀀1. We divide by x + 1 to obtain
the equation of a line.
y + 2 = x 􀀀 1
y = x 􀀀 3
Now we collect the solutions we have found.
f(􀀀1; y) : y 6= 2g [ f(x; x 􀀀 3) : x 6= 1; 5g
The solutions are depicted in Figure /refg not a line.
16
-6 -4 -2 2 4 6
-6
-4
-2
2
4
6
Figure 1.12: The solutions of x+1
y􀀀2 = x2􀀀1
y2􀀀4 .
Solution 1.3
The denominator is nonzero for all x 2 R. Since we don't have any division by zero problems, the domain of the
function is R. For x 2 R,
0 <
1
x2 + 2
 2:
Consider
y =
1
x2 + 2
: (1.1)
For any y 2 (0 : : : 1=2], there is at least one value of x that satises Equation 1.1.
x2 + 2 =
1
y
x = 
r
1
y
􀀀 2
Thus the range of the function is (0 : : : 1=2]
17
Solution 1.4
Let c denote degrees Celsius and f denote degrees Fahrenheit. The line passes through the points (f; c) = (32; 0) and
(f; c) = (212; 100). The x-intercept is f = 32. We calculate the slope of the line.
slope =
100 􀀀 0
212 􀀀 32
=
100
180
=
5
9
The relationship between fahrenheit and celcius is
c =
5
9
(f 􀀀 32):
Solution 1.5
We plot the various transformations of f(x).
Solution 1.6
The formula for geometric growth/decay is x(t) = x0rt, where r is the rate. Let t = 0 coincide with 6:00 pm. We
determine x0.
x(0) = 109 = x0

11
10
0
= x0
x0 = 109
At 7:00 pm the number of bacteria is
109

11
10
60
=
1160
1051
 3:04  1011
At 3:00 pm the number of bacteria was
109

11
10
􀀀180
=
10189
11180
 35:4
18
Figure 1.13: Graphs of f(􀀀x), f(x + 3), f(3 􀀀 x) + 2, and f􀀀1(x).
Solution 1.7
We write p(x) and q(x) as general quadratic polynomials.
f(x) =
p(x)
q(x)
=
ax2 + bx + c
x2 + x + 
We will use the properties of the function to solve for the unknown parameters.
19
Note that p(x) and q(x) appear as a ratio, they are determined only up to a multiplicative constant. We may take
the leading coecient of q(x) to be unity.
f(x) =
p(x)
q(x)
=
ax2 + bx + c
x2 + x + 
f(x) has a second order zero at x = 0. This means that p(x) has a second order zero there and that  6= 0.
f(x) =
ax2
x2 + x + 
We note that f(x) ! 2 as x ! 1. This determines the parameter a.
lim
x!1
f(x) = lim
x!1
ax2
x2 + x + 
= lim
x!1
2ax
2x + 
= lim
x!1
2a
2
= a
f(x) =
2x2
x2 + x + 
Now we use the fact that f(x) is even to conclude that q(x) is even and thus  = 0.
f(x) =
2x2
x2 + 
Finally, we use that f(1) = 1 to determine .
f(x) =
2x2
x2 + 1
20
Solution 1.8
Consider the polynomial
p(x) = (x + 2)40(x 􀀀 1)30(x 􀀀 )30:
It is of degree 100. Since the factors only vanish at x = 􀀀2; 1; , p(x) only vanishes there. Since factors are nonnegative,
the polynomial is non-negative.
21
Chapter 2
Vectors
2.1 Vectors
2.1.1 Scalars and Vectors
A vector is a quantity having both a magnitude and a direction. Examples of vector quantities are velocity, force
and position. One can represent a vector in n-dimensional space with an arrow whose initial point is at the origin,
(Figure 2.1). The magnitude is the length of the vector. Typographically, variables representing vectors are often
written in capital letters, bold face or with a vector over-line, A; a;~a. The magnitude of a vector is denoted jaj.
A scalar has only a magnitude. Examples of scalar quantities are mass, time and speed.
Vector Algebra. Two vectors are equal if they have the same magnitude and direction. The negative of a vector,
denoted 􀀀a, is a vector of the same magnitude as a but in the opposite direction. We add two vectors a and b by
placing the tail of b at the head of a and dening a + b to be the vector with tail at the origin and head at the head
of b. (See Figure 2.2.)
The dierence, a 􀀀 b, is dened as the sum of a and the negative of b, a + (􀀀b). The result of multiplying a by
a scalar  is a vector of magnitude jj jaj with the same/opposite direction if  is positive/negative. (See Figure 2.2.)
22
x
z
y
Figure 2.1: Graphical Representation of a Vector in Three Dimensions
a+b
a
b
-a
a
2a
Figure 2.2: Vector Arithmetic
Here are the properties of adding vectors and multiplying them by a scalar. They are evident from geometric
considerations.
a + b = b + a a = a commutative laws
(a + b) + c = a + (b + c) (a) = ()a associative laws
(a + b) = a + b ( + )a = a + a distributive laws
23
Zero and Unit Vectors. The additive identity element for vectors is the zero vector or null vector. This is a vector
of magnitude zero which is denoted as 0. A unit vector is a vector of magnitude one. If a is nonzero then a=jaj is a
unit vector in the direction of a. Unit vectors are often denoted with a caret over-line, ^n.
Rectangular Unit Vectors. In n dimensional Cartesian space, Rn, the unit vectors in the directions of the
coordinates axes are e1; : : : en. These are called the rectangular unit vectors. To cut down on subscripts, the unit
vectors in three dimensional space are often denoted with i, j and k. (Figure 2.3).
x
z
y
j
k
i
Figure 2.3: Rectangular Unit Vectors
Components of a Vector. Consider a vector a with tail at the origin and head having the Cartesian coordinates
(a1; : : : ; an). We can represent this vector as the sum of n rectangular component vectors, a = a1e1 +


+ anen.
(See Figure 2.4.) Another notation for the vector a is ha1; : : : ; ani. By the Pythagorean theorem, the magnitude of
the vector a is jaj =
p
a21
+


+ a2
n.
24
x
z
y
a
a
a
1
3
i
k
a2 j
Figure 2.4: Components of a Vector
2.1.2 The Kronecker Delta and Einstein Summation Convention
The Kronecker Delta tensor is dened
ij =
(
1 if i = j;
0 if i 6= j:
This notation will be useful in our work with vectors.
Consider writing a vector in terms of its rectangular components. Instead of using ellipses: a = a1e1+


+anen, we
could write the expression as a sum: a =
Pn
i=1 aiei. We can shorten this notation by leaving out the sum: a = aiei,
where it is understood that whenever an index is repeated in a term we sum over that index from 1 to n. This is the
Einstein summation convention. A repeated index is called a summation index or a dummy index. Other indices can
take any value from 1 to n and are called free indices.
25
Example 2.1.1 Consider the matrix equation: A
x = b. We can write out the matrix and vectors explicitly.
0
B@
a11


a1n
...
. . .
...
an1


ann
1
CA
0
B@
x1
...
xn
1
CA
=
0
B@
b1
...
bn
1
CA
This takes much less space when we use the summation convention.
aijxj = bi
Here j is a summation index and i is a free index.
2.1.3 The Dot and Cross Product
Dot Product. The dot product or scalar product of two vectors is dened,
a
b  jajjbj cos ;
where  is the angle from a to b. From this denition one can derive the following properties:
 a
b = b
a, commutative.
 (a
b) = (a)
b = a
(b), associativity of scalar multiplication.
 a
(b + c) = a
b + a
c, distributive.
 eiej = ij . In three dimension, this is
i
i = j
j = k
k = 1; i
j = j
k = k
i = 0:
 a
b = aibi  a1b1 +


+ anbn, dot product in terms of rectangular components.
 If a
b = 0 then either a and b are orthogonal, (perpendicular), or one of a and b are zero.
26
The Angle Between Two Vectors. We can use the dot product to nd the angle between two vectors, a and
b. From the denition of the dot product,
a
b = jajjbj cos :
If the vectors are nonzero, then
 = arccos

a
b
jajjbj

:
Example 2.1.2 What is the angle between i and i + j?
 = arccos

i
(i + j)
jijji + jj

= arccos

1
p
2

=

4
:
Parametric Equation of a Line. Consider a line that passes through the point a and is parallel to the vector t,
(tangent). A parametric equation of the line is
x = a + ut; u 2 R:
Implicit Equation of a Line. Consider a line that passes through the point a and is normal, (orthogonal, perpendicular),
to the vector n. All the lines that are normal to n have the property that x
n is a constant, where x is
any point on the line. (See Figure 2.5.) x
n = 0 is the line that is normal to n and passes through the origin. The
line that is normal to n and passes through the point a is
x
n = a
n:
27
=0
=1 =a n
n a
=-1
x n
x n
x n
x n
Figure 2.5: Equation for a Line
The normal to a line determines an orientation of the line. The normal points in the direction that is above the
line. A point b is (above/on/below) the line if (b 􀀀 a)
n is (positive/zero/negative). The signed distance of a point
b from the line x
n = a
n is
(b 􀀀 a)

n
jnj
:
Implicit Equation of a Hyperplane. A hyperplane in Rn is an n􀀀1 dimensional \sheet" which passes through
a given point and is normal to a given direction. In R3 we call this a plane. Consider a hyperplane that passes through
the point a and is normal to the vector n. All the hyperplanes that are normal to n have the property that x
n is a
constant, where x is any point in the hyperplane. x
n = 0 is the hyperplane that is normal to n and passes through
the origin. The hyperplane that is normal to n and passes through the point a is
x
n = a
n:
The normal determines an orientation of the hyperplane. The normal points in the direction that is above the
hyperplane. A point b is (above/on/below) the hyperplane if (b 􀀀 a)
n is (positive/zero/negative). The signed
28
distance of a point b from the hyperplane x
n = a
n is
(b 􀀀 a)

n
jnj
:
Right and Left-Handed Coordinate Systems. Consider a rectangular coordinate system in two dimensions.
Angles are measured from the positive x axis in the direction of the positive y axis. There are two ways of labeling the
axes. (See Figure 2.6.) In one the angle increases in the counterclockwise direction and in the other the angle increases
in the clockwise direction. The former is the familiar Cartesian coordinate system.
x y
y x
q
q
Figure 2.6: There are Two Ways of Labeling the Axes in Two Dimensions.
There are also two ways of labeling the axes in a three-dimensional rectangular coordinate system. These are called
right-handed and left-handed coordinate systems. See Figure 2.7. Any other labelling of the axes could be rotated into
one of these congurations. The right-handed system is the one that is used by default. If you put your right thumb in
the direction of the z axis in a right-handed coordinate system, then your ngers curl in the direction from the x axis
to the y axis.
Cross Product. The cross product or vector product is dened,
a  b = jajjbj sin  n;
where  is the angle from a to b and n is a unit vector that is orthogonal to a and b and in the direction such that a,
b and n form a right-handed system.
29
x
z
j y
i
k
z
k
j
i
y
x
Figure 2.7: Right and Left Handed Coordinate Systems
You can visualize the direction of a  b by applying the right hand rule. Curl the ngers of your right hand in the
direction from a to b. Your thumb points in the direction of a  b. Warning: Unless you are a lefty, get in the habit
of putting down your pencil before applying the right hand rule.
The dot and cross products behave a little dierently. First note that unlike the dot product, the cross product is not
commutative. The magnitudes of a  b and b  a are the same, but their directions are opposite. (See Figure 2.8.)
a
b
b a
a b
Figure 2.8: The Cross Product is Anti-Commutative.
30
Let
a  b = jajjbj sin  n and b  a = jbjjaj sin  m:
The angle from a to b is the same as the angle from b to a. Since fa; b; ng and fb; a;mg are right-handed systems,
m points in the opposite direction as n. Since a  b = 􀀀b  a we say that the cross product is anti-commutative.
Next we note that since
ja  bj = jajjbj sin ;
the magnitude of a  b is the area of the parallelogram dened by the two vectors. (See Figure 2.9.) The area of the
triangle dened by two vectors is then 1
2
ja  bj.
b
sin
b
b
a
q
a
Figure 2.9: The Parallelogram and the Triangle Dened by Two Vectors
From the denition of the cross product, one can derive the following properties:
 a  b = 􀀀b  a, anti-commutative.
 (a  b) = (a)  b = a  (b), associativity of scalar multiplication.
 a  (b + c) = a  b + a  c, distributive.
 (a  b)  c 6= a  (b  c). The cross product is not associative.
 i  i = j  j = k  k = 0.
31
 i  j = k, j  k = i, k  i = j.

a  b = (a2b3 􀀀 a3b2)i + (a3b1 􀀀 a1b3)j + (a1b2 􀀀 a2b1)k =

i j k
a1 a2 a3
b1 b2 b3

;
cross product in terms of rectangular components.
 If a
b = 0 then either a and b are parallel or one of a or b is zero.
Scalar Triple Product. Consider the volume of the parallelopiped dened by three vectors. (See Figure 2.10.)
The area of the base is jjbjjcj sin j, where  is the angle between b and c. The height is jaj cos , where  is the angle
between b  c and a. Thus the volume of the parallelopiped is jajjbjjcj sin  cos .
f
q
b c
a
b
c
Figure 2.10: The Parallelopiped Dened by Three Vectors
Note that
ja
(b  c)j = ja
(jbjjcj sin  n)j
= jjajjbjjcj sin  cos j :
32
Thus ja
(b  c)j is the volume of the parallelopiped. a
(bc) is the volume or the negative of the volume depending
on whether fa; b; cg is a right or left-handed system.
Note that parentheses are unnecessary in a
bc. There is only one way to interpret the expression. If you did the
dot product rst then you would be left with the cross product of a scalar and a vector which is meaningless. a
bc
is called the scalar triple product.
Plane Dened by Three Points. Three points which are not collinear dene a plane. Consider a plane that
passes through the three points a, b and c. One way of expressing that the point x lies in the plane is that the vectors
x 􀀀 a, b 􀀀 a and c 􀀀 a are coplanar. (See Figure 2.11.) If the vectors are coplanar, then the parallelopiped dened by
these three vectors will have zero volume. We can express this in an equation using the scalar triple product,
(x 􀀀 a)
(b 􀀀 a)  (c 􀀀 a) = 0:
b
c
x
a
Figure 2.11: Three Points Dene a Plane.
2.2 Sets of Vectors in n Dimensions
Orthogonality. Consider two n-dimensional vectors
x = (x1; x2; : : : ; xn); y = (y1; y2; : : : ; yn):
33
The inner product of these vectors can be dened
hxjyi  x
y =
Xn
i=1
xiyi:
The vectors are orthogonal if x
y = 0. The norm of a vector is the length of the vector generalized to n dimensions.
kxk =
p
x
x
Consider a set of vectors
fx1; x2; : : : ; xmg:
If each pair of vectors in the set is orthogonal, then the set is orthogonal.
xi
xj = 0 if i 6= j
If in addition each vector in the set has norm 1, then the set is orthonormal.
xi
xj = ij =
(
1 if i = j
0 if i 6= j
Here ij is known as the Kronecker delta function.
Completeness. A set of n, n-dimensional vectors
fx1; x2; : : : ; xng
is complete if any n-dimensional vector can be written as a linear combination of the vectors in the set. That is, any
vector y can be written
y =
Xn
i=1
cixi:
34
Taking the inner product of each side of this equation with xm,
y
xm =
Xn
i=1
cixi
!

xm
=
Xn
i=1
cixi
xm
= cmxm
xm
cm =
y
xm
kxmk2
Thus y has the expansion
y =
Xn
i=1
y
xi
kxik2xi:
If in addition the set is orthonormal, then
y =
Xn
i=1
(y
xi)xi:
35
2.3 Exercises
The Dot and Cross Product
Exercise 2.1
Prove the distributive law for the dot product,
a
(b + c) = a
b + a
c:
Exercise 2.2
Prove that
a
b = aibi  a1b1 +


+ anbn:
Exercise 2.3
What is the angle between the vectors i + j and i + 3j?
Exercise 2.4
Prove the distributive law for the cross product,
a  (b + c) = a  b + a  b:
Exercise 2.5
Show that
a  b =

i j k
a1 a2 a3
b1 b2 b3

Exercise 2.6
What is the area of the quadrilateral with vertices at (1; 1), (4; 2), (3; 7) and (2; 3)?
Exercise 2.7
What is the volume of the tetrahedron with vertices at (1; 1; 0), (3; 2; 1), (2; 4; 1) and (1; 2; 5)?
36
Exercise 2.8
What is the equation of the plane that passes through the points (1; 2; 3), (2; 3; 1) and (3; 1; 2)? What is the distance
from the point (2; 3; 5) to the plane?
37
2.4 Hints
The Dot and Cross Product
Hint 2.1
First prove the distributive law when the rst vector is of unit length,
n
(b + c) = n
b + n
c:
Then all the quantities in the equation are projections onto the unit vector n and you can use geometry.
Hint 2.2
First prove that the dot product of a rectangular unit vector with itself is one and the dot product of two distinct
rectangular unit vectors is zero. Then write a and b in rectangular components and use the distributive law.
Hint 2.3
Use a
b = jajjbj cos .
Hint 2.4
First consider the case that both b and c are orthogonal to a. Prove the distributive law in this case from geometric
considerations.
Next consider two arbitrary vectors a and b. We can write b = b? + bk where b? is orthogonal to a and bk is
parallel to a. Show that
a  b = a  b?:
Finally prove the distributive law for arbitrary b and c.
Hint 2.5
Write the vectors in their rectangular components and use,
i  j = k; j  k = i; k  i = j;
and,
i  i = j  j = k  k = 0:
38
Hint 2.6
The quadrilateral is composed of two triangles. The area of a triangle dened by the two vectors a and b is 1
2
ja
bj.
Hint 2.7
Justify that the area of a tetrahedron determined by three vectors is one sixth the area of the parallelogram determined
by those three vectors. The area of a parallelogram determined by three vectors is the magnitude of the scalar triple
product of the vectors: a
b  c.
Hint 2.8
The equation of a line that is orthogonal to a and passes through the point b is a
x = a
b. The distance of a point
c from the plane is (c 􀀀 b)

a
jaj

39
2.5 Solutions
The Dot and Cross Product
Solution 2.1
First we prove the distributive law when the rst vector is of unit length, i.e.,
n
(b + c) = n
b + n
c: (2.1)
From Figure 2.12 we see that the projection of the vector b+c onto n is equal to the sum of the projections b
n and
c
n.
b
c
n b
n c
b+c
n
n (b+c)
Figure 2.12: The Distributive Law for the Dot Product
Now we extend the result to the case when the rst vector has arbitrary length. We dene a = jajn and multiply
Equation 2.1 by the scalar, jaj.
jajn
(b + c) = jajn
b + jajn
c
a
(b + c) = a
b + a
c:
Solution 2.2
First note that
ei
ei = jeijjeij cos(0) = 1:
40
Then note that that dot product of any two distinct rectangular unit vectors is zero because they are orthogonal. Now
we write a and b in terms of their rectangular components and use the distributive law.
a
b = aiei
bjej
= aibjei
ej
= aibjij
= aibi
Solution 2.3
Since a
b = jajjbj cos , we have
 = arccos

a
b
jajjbj

when a and b are nonzero.
 = arccos

(i + j)
(i + 3j)
ji + jjji + 3jj

= arccos

4
p
2
p
10

= arccos
2
p
5
5
!
 0:463648
Solution 2.4
First consider the case that both b and c are orthogonal to a. b + c is the diagonal of the parallelogram dened by
b and c, (see Figure 2.13). Since a is orthogonal to each of these vectors, taking the cross product of a with these
vectors has the eect of rotating the vectors through =2 radians about a and multiplying their length by jaj. Note
that a(b+c) is the diagonal of the parallelogram dened by ab and ac. Thus we see that the distributive law
holds when a is orthogonal to both b and c,
a  (b + c) = a  b + a  c:
Now consider two arbitrary vectors a and b. We can write b = b? + bk where b? is orthogonal to a and bk is
parallel to a, (see Figure 2.14).
By the denition of the cross product,
a  b = jajjbj sin  n:
41
b
b+c c
a c
a
a b
a (b+c)
Figure 2.13: The Distributive Law for the Cross Product
a
b
b
q
b
Figure 2.14: The Vector b Written as a Sum of Components Orthogonal and Parallel to a
Note that
jb?j = jbj sin ;
42
and that a  b? is a vector in the same direction as a  b. Thus we see that
a  b = jajjbj sin  n
= jaj(sin jbj)n
= jajjb?jn = jajjb?j sin(=2)n
a  b = a  b?:
Now we are prepared to prove the distributive law for arbitrary b and c.
a  (b + c) = a  (b? + bk + c? + ck)
= a  ((b + c)? + (b + c)k)
= a  ((b + c)?)
= a  b? + a  c?
= a  b + a  c
a  (b + c) = a  b + a  c
Solution 2.5
We know that
i  j = k; j  k = i; k  i = j;
and that
i  i = j  j = k  k = 0:
Now we write a and b in terms of their rectangular components and use the distributive law to expand the cross
product.
a  b = (a1i + a2j + a3k)  (b1i + b2j + b3k)
= a1i  (b1i + b2j + b3k) + a2j  (b1i + b2j + b3k) + a3k  (b1i + b2j + b3k)
= a1b2k + a1b3(􀀀j) + a2b1(􀀀k) + a2b3i + a3b1j + a3b2(􀀀i)
= (a2b3 􀀀 a3b2)i 􀀀 (a1b3 􀀀 a3b1)j + (a1b2 􀀀 a2b1)k
43
Next we evaluate the determinant.

i j k
a1 a2 a3
b1 b2 b3

= i

a2 a3
b2 b3

􀀀 j

a1 a3
b1 b3

+ k

a1 a2
b1 b2

= (a2b3 􀀀 a3b2)i 􀀀 (a1b3 􀀀 a3b1)j + (a1b2 􀀀 a2b1)k
Thus we see that,
a  b =

i j k
a1 a2 a3
b1 b2 b3

Solution 2.6
The area area of the quadrilateral is the area of two triangles. The rst triangle is dened by the vector from (1; 1) to
(4; 2) and the vector from (1; 1) to (2; 3). The second triangle is dened by the vector from (3; 7) to (4; 2) and the
vector from (3; 7) to (2; 3). (See Figure 2.15.) The area of a triangle dened by the two vectors a and b is 1
2
ja
bj.
The area of the quadrilateral is then,
1
2
j(3i + j)
(i + 2j)j +
1
2
j(i 􀀀 5j)
(􀀀i 􀀀 4j)j =
1
2
(5) +
1
2
(19) = 12:
Solution 2.7
The tetrahedron is determined by the three vectors with tail at (1; 1; 0) and heads at (3; 2; 1), (2; 4; 1) and (1; 2; 5).
These are h2; 1; 1i, h1; 3; 1i and h0; 1; 5i. The area of the tetrahedron is one sixth the area of the parallelogram
determined by these vectors. (This is because the area of a pyramid is 1
3 (base)(height). The base of the tetrahedron is
half the area of the parallelogram and the heights are the same. 1
2
1
3 = 1
6 ) Thus the area of a tetrahedron determined
by three vectors is 1
6
ja
b  cj. The area of the tetrahedron is
1
6
jh2; 1; 1i
h1; 3; 1i  h1; 2; 5ij =
1
6
jh2; 1; 1i
h13;􀀀4;􀀀1ij =
7
2
44
x
y (3,7)
(4,2)
(2,3)
(1,1)
Figure 2.15: Quadrilateral
Solution 2.8
The two vectors with tails at (1; 2; 3) and heads at (2; 3; 1) and (3; 1; 2) are parallel to the plane. Taking the cross
product of these two vectors gives us a vector that is orthogonal to the plane.
h1; 1;􀀀2i  h2;􀀀1;􀀀1i = h􀀀3;􀀀3;􀀀3i
We see that the plane is orthogonal to the vector h1; 1; 1i and passes through the point (1; 2; 3). The equation of the
plane is
h1; 1; 1i
hx; y; zi = h1; 1; 1i
h1; 2; 3i;
x + y + z = 6:
Consider the vector with tail at (1; 2; 3) and head at (2; 3; 5). The magnitude of the dot product of this vector with
the unit normal vector gives the distance from the plane.

h1; 1; 2i

h1; 1; 1i
jh1; 1; 1ij

=
4
p
3
=
4
p
3
3
45