Vector Calculus Using Mathematica Second Edition

Vector Calculus Using Mathematica®

Second Edition

With 165 fully solved examples, 255 figures and 9 animations

Steven Tan

Copyright

Vector Calculus Using Mathematica®

Copyright © 2020, 2018 Steven Tan

Second edition July 2020

First edition April 2018

Published by Lulu Press, Inc.

ISBN  978-1-71675-280-3

The programs and applications presented in this book have been included for their instructional value. They have been tested with care but are not guaranteed for any particular purpose. The publisher does not offer any warranties or representations, nor does it accept any liabilities with respect to the programs or applications.

All rights reserved. Apart from any fair dealing for the purposes of research or private study or criticism or review, no part of this publication may be reproduced, stored or transmitted in any form or by any means, without the prior permission in writing of the publisher.

Dedication

To Svetlana

Preface to the Second Edition

The second edition is an addition of the first. Two new chapters on line integrals, Green's Theorem, Stokes's Theorem and Gauss's Theorem have been added.

The author would like to thank everyone who pointed out mistakes in the first edition. We hope we have found and corrected these, and have not introduced any more with the new material.

Preface to the First Edition

As far as the laws of mathematics refer to reality, they are not certain;

and as far as they are certain, they do not refer to reality.

(Illuminated Geometry, Sidelights on Relativity by A. Einstein)

This book is intended for an undergraduate course in vector calculus taken by students majoring in science, or engineering. There are many standard textbooks on vector calculus, however, this book takes rather different approach.

First of all, this book is different from most textbooks because it is the author's attempt to introduce vector calculus with the aid of Mathematica® computer algebra system to represent them and to calculate with them. It is written with and for Mathematica® version 11. However, most illustrations are backward compatible with earlier versions of Mathematica® or have equivalent representations.

Secondly, the unique features of this book, which set it apart from the existing textbooks, are the large number of illustrative examples. It is the author's opinion a novice in science or engineering needs to see a lot of examples in which mathematics is used to be able to speak the language. Throughout the book these examples and all illustrations can be replicated and used to learn and discover vector calculus in a new and exciting way. Since we cover the problems from the core courses, the reader can practice with our solutions, and then modify our solutions to solve the particular problems assigned. This should help the reader move up the problem solving skills and to use Mathematica® to visualize the results and to develop a deeper intuitive understanding. Usually, visualization provides much more insight than the formulas themselves.

This book covers vector calculus in one, two and three variables. The prerequisites are the standard courses in calculus. A knowledge of computer programming would be beneficial but not essential. From the point of view of mathematicians, this book is less rigor because the author has put less emphasis on proofs but more on examples and illustrations to clarify concepts.

It is important to note that this book should not be thought of as a substitute for the text in a chapter in a standard introductory textbook. Although there is an introduction to each chapter where the basics are presented, this introduction is brief. This book is therefore designed to be used in conjunction with any standard textbook. You could think of this book as supplementing a textbook by providing examples using Mathematica®. Or you could think of a textbook as supplementing this book by providing additional conceptual and theoretical background; and proofs of theorems.

We assume the reader has some familiarity with Mathematica®, so we could focus directly on vector calculus. This book does not discuss programming in Mathematica® nor does it teach all the principles and techniques of applications using Mathematica®. However, Appendix A provides a basic introduction to Mathematica®. New users will find that the materials in this appendix enable them to become familiar with Mathematica® within a few hours. The reader can learn the essentials of Mathematica® through examples described in the book. Mathematica® commands and techniques are introduced as the need arises.

Although extreme care was taken to correct all the misprints, it is very unlikely that I have been able to catch all of them. I shall be most grateful to those readers kind enough to bring to my attention any remaining mistakes, typographical or otherwise. Please feel free to contact me at:

[email protected]

Steven Tan

Table of Contents

Preface

Chapter 1. Vectors

1.1 Algebraic Properties of Vectors

1.2 Geometric Properties of Vectors

1.3 Parametric Equations of Curves

1.3.1 The right circular helix of unit radius

1.3.2 A three-dimensional astroid

1.3.3 A cycloid

1.3.4 The involute of a circle

1.4 The Dot Product

1.5 The Cross Product of Vectors in

1.5.1 The Area of a Parallelogram

1.5.2 The Volume of a Parallelepiped

1.6 Equations for Planes

1.6.1 Coordinate Equation of Planes

1.6.2 Parametric Equations of Planes

1.7 Distance Examples

1.7.1 Distance between a point and a line

1.7.2 Distance between a point and a plane

1.7.3 Distance between parallel planes

1.7.4 Distance between two skew lines

1.8 Curvilinear Coordinates

1.8.1 Polar Coordinates

1.8.2 Cylindrical Coordinates

1.8.3 Spherical Coordinates

Chapter 2. Curves

2.1 Parametric Curves

2.1.1 Reparametrization

2.1.1.1 Reparametrization by arc length

2.1.2 The Frenet Formulas

2.2 Osculating circle to plane curves

2.3 Some applications

2.3.1 Equations of motion in curvilinear coordinates

2.3.1.1 Polar coordinates

2.3.1.2 Cylindrical coordinates

2.3.1.3 Spherical coordinates

2.3.2 Kepler's Laws of Planetary Motion

2.3.3 The Lorentz force

Chapter 3. Differentiation in Several Variables

3.1 Visualizing Functions

3.1.1 Functions of Two Variables

3.1.2 Functions of Three Variables

3.1.3 Parametric Surfaces

3.2 The Derivative

3.2.1 Directional Derivatives

Chapter 4. Surfaces

4.1 Coordinate Curves, Tangent Planes, and Normal Vectors

4.2 Surface Integrals

4.2.1 Area of a Parametrized Surface

4.2.2 Scalar Surface Integrals

4.2.3 Vector Surface Integrals

4.3 Curvatures of Surfaces

4.4 Some Classification of Surfaces

4.4.1 Monge Patch

4.4.2 Quadric Surface

4.4.3 Cylindrical Surface

4.4.4 Ruled Surface

4.4.5 Surface of Revolution

Chapter 5. Optimization in Several Variables

5.1 Second Derivative Test

5.2 Global Extrema

5.3 Lagrange Multipliers: Constrained Extrema

Chapter 6. Multiple Integration

6.1 Integration in Two Variables

6.1.1 The Integral over a Rectangle

6.1.2 Double Integrals over General Regions

6.1.3 Changing the Order of Integration

6.2 Triple Integrals

6.2.1 The Integral over a Box

6.2.2 Triple Integrals over Arbitrary Solid Regions

6.3 Change of Variables in Double Integrals

6.3.1 Double Integrals in Polar Coordinates

6.3.2 General Change of Variables in Double Integrals

6.4 Change of Variables in Triple Integrals

6.4.1 Triple Integrals in Cylindrical Coordinates

6.4.2 Triple Integrals in Spherical Coordinates

Chapter 7. Line Integrals and Green's Theorem

7.1 Line Integrals

7.1.1 Scalar Line Integrals

7.1.2 Vector Line Integrals

7.2 Green's Theorem

Chapter 8. Stokes's and Gauss's Theorems

8.1 Stokes's Theorem

8.2 Gauss's Theorems

Appendix A. Introduction to Mathematica

A.1 Executing Simple Expressions

A.2 Defining a function

A.3 Lists

A.4 Plotting Graphs

A.5 Calculus

A.6 Differential Equations

A.7 Modules

Bibliography

Chapter 1. Vectors

Initialization

This section is reserved for Mathematica notebook initialization.

At the start of each chapter, we shall clear all values and definitions of any previously defined symbols using Clear command. Here is also the place where we set some default options. Most Mathematica modules are defined in the texts when the needs arise.

Clear[Global`*]

SetOptions[ParametricPlot,PlotStyle->AbsoluteThickness[2]];

SetOptions[ParametricPlot3D,PlotStyle->AbsoluteThickness[2]];

SetOptions[Graphics,BaseStyle->AbsoluteThickness[2]];

SetOptions[Graphics3D,BaseStyle->AbsoluteThickness[2]];

Using SetOptions command we set specified default options for symbols that we are going to use thereafter. Unless overwrite it explicitly the options for these commands are set according to the values in SetOptions. For example, the ParametricPlot command's PlotStyle option is set to AbsoluteThickness[2] by default.

1.1 Algebraic Properties of Vectors

We begin with some reviews of the notion of a vector.

Definition 1.1

A vector in is an ordered pair of real numbers. A vector in is an ordered triple of real numbers. Similarly, a vector in is an ordered n-tuple of real numbers.

In Mathematica, vectors are represented as lists of numbers or variables encased in braces.

vector2D={v1,v2}

vector3D={v1,v2,v3}

Show that addition of vectors is commutative, that is, A + B = B + A.

a={a1,a2,a3}

b={b1,b2,b3}

a+b==b+a

Show that the addition of vectors is associative, that is, A + (B + C) = (A + B) + C.

c={c1,c2,c3}

a+(b+c)==(a+b)+c

If A, B, and C are vectors, and m and n are scalars, then

1. A + B = B + A Commutative law for addition

2. A + (B + C) = (A + B) + C Associative law for addition

3. m(nA) = (m n)A = n(mA) Associative law for scalar multiplication

4. (m + n)A = mA + nA Distributive law for scalars

5. m(A + B) = mA + mB Distributive law for vectors

1.2 Geometric Properties of Vectors

A vector can be visualized in or as an arrow that begins at the origin and ends at a point. Such a depiction is often called the position vector of the point or in or , respectively.

(* A vector in *)

Graphics[{Arrowheads[0.1],Arrow[{{0,0},{2,1}}]},Axes->True]

Figure 1.1. The vector a=(2,1) in .

(* A vector in *)

Graphics3D[{Arrowheads[0.1],Arrow[{{0,0,0},{3,3,3}}]},Axes->True]

Figure 1.2. The vector a=(3,3,3) in .

Two vectors are equivalent as long as they are parallel and have the same lengths. Thus, we may represent the vector by an arrow with its tail at the origin or with its tail at any other point.

origin={0,0};

a={2,1};

b={1,2};

Graphics[{Blue,Arrowheads[.08],Arrow[{origin,a}],Arrow[{origin,b}],Arrow[{origin,a+b}],{Dashed,Arrow[{a,a+b}]},{Dashed,Arrow[{b,a+b}]},Text[Style[a,Bold,FontSize->14],{1,0.3}],Text[Style[b,Bold,FontSize->14],{0.5,1.3}],Text[Style[a + b,Bold,FontSize->14],{1.6,1.3}]},Axes->True]

Figure 1.3. The vector a+b is represented by the arrow that begins at the common initial point of a and b and runs along the diagonal of the parallelogram determined by a and b.

1.3 Parametric Equations of Curves

A curve in is the set of points (x,y) whose coordinates are given by a function of a parameter t, such as

(1.1)           

Similarly, a curve in is the set of points (x,y,z) whose coordinates are given by a function of a parameter t

(1.2)           

Example 1.1. The vector parametric equations for a line in through the point and parallel to are given by

where

Example 1.2. Similarly, the vector parametric equations for a line in through the point and parallel to are

where

In vector notation, this can be written as

where

Definition 1.2

Let and be two nonzero, nonparallel vectors in . The vector parametric equations for a line in through the point b and parallel to a is defined as

(1.3)

where and .

Example 1.3. Write a set of parametric equations for the line through the points (5,-3,4) and (0,1,9). Draw the line.

Solution:

Clear[a,b]

a={0,1,9}-{5,-3,4}

b={5,-3,4}

(* The equation of the line through b and parallel to a *)

line=a t+b

Thus, the set of parametric equation for the line is

In vector notation, it is written as

r(t)=(-5t+5,4t-3,5t+4).

Note that we can use either the point (5,-3,4) or (0,1,9) as the through-point for b.

Module[{plot},plot[1]=ParametricPlot3D[line,{t,-1.5,1.5},PlotStyle->Hue[0.7],Boxed->False,AxesOrigin->{0,0,0},AxesLabel->{x,y,z}];

plot[2]=Graphics3D[{{Dashed,AbsoluteThickness[2],Arrowheads[0.04],Arrow[{{0,0,0},b}]},{Red,AbsoluteThickness[2],Arrowheads[0.04],Arrow[{b,b+0.5a}]}}];

plot[3]=Graphics3D[{{Text[Style[b,Bold],{3.5,-3,3.3}]},{Text[Style[a,Bold],{5,-3,5.5}]},{Text[Style[(5, -3, 4),Bold],{7.1,-3,4.5}]},{Text[Style[(0, 1, 9),Bold],{4,-3,10}]}}];

plot[4]=Graphics3D[{{Red,PointSize[0.025],Point[b]},{PointSize[0.025],Point[{0,1,9}]}}];

Show[Array[plot,4],ViewPoint->{5,-10,2}]

]

Figure 1.4. The line in space parametrized by the equation r(t)=(-5t+5,4t-3,5t+4).

1.3.1 The right circular helix of unit radius

The parametric equations for a right circular helix of radius 1 are defined as

ParametricPlot3D[{Cos[t],Sin[t],t},{t,-5,5},PlotStyle->Hue[0.7],AspectRatio->2]

Figure 1.5. The right circular helix of radius 1.

1.3.2 A three-dimensional astroid

The parametric equations for a three-dimensional astroid are given by

ParametricPlot3D[{Cos[t]^3,Sin[t]^3,Cos[2t]}//Evaluate,{t,0,2π},PlotStyle->Hue[0.7]]

Figure 1.6. A three-dimensional astroid.

1.3.3 A cycloid

If a wheel rolls along a straight line at constant speed without slipping, a point on the circumference traces a curve called a cycloid (see Tan, S. Handbook of Famous Plane Curves Using Mathematica, Chapter 2.13 Cycloid).

The parametric equations for a cycloid are given by

ParametricPlot[{t-Sin[t],1-Cos[t]}//Evaluate,{t,0,4π},PlotStyle->Hue[0.7]]

Figure 1.7. The cycloid for a=1.

Module[{x,y,cycl},

x[t_]:=t-Sin[t];

y[t_]:=1-Cos[t];

cycl[u_]:=ParametricPlot[{{x[t],y[t]},{u+Cos[t],1+Sin[t]}},{t,0,6 π},Epilog->{{PointSize[0.01],Black,Point[{u,1}],PointSize[0.015],Red,Point[{x[u],y[u]}]},Line[{{u,1},{x[u],y[u]}}]},PlotRange->{{-1,20},{-2,3}},ImageSize->550];

Animate[cycl[u],{u,0,6π},AnimationRunning->False]

]

Animation 1.1. The locus of a point on the rim of a circle of radius 1 rolling along a straight line generates the cycloid (t-sin t,1-cos t).

1.3.4 The involute of a circle

The involute of a circle is formed by unwinding a taut string which has been wrapped around a circle (see Tan, S. Handbook of Famous Plane Curves Using Mathematica, Chapter 2.31 Involute of a Circle).

The parametric equations are

ParametricPlot[{{Cos[t],Sin[t]},{Cos[t]+t Sin[t],Sin[t]-t Cos[t]}}//Evaluate,{t,0,2π},PlotStyle->Table[Hue[0.7i],{i,0,1}]]

Figure 1.8. The involute of a circle.

1.4 The Dot Product

Definition 1.3

Let and be two vectors in . The dot product of a and b is defined as

In , the analogous definition is

(1.4)

where and .

In Mathematica, there are two ways to compute the dot product, that is,

1. Using the period key between two vectors

a.b

2. Using Dot command

Dot[a,b]

Properties of dot products.

If and , then

1. a · a>=0

    a · a=0 if and only if a=0.

2. a · b=b · a

3. a · (b+c)= a · b + a · c

4. k (a · b)=(k a) · b=a · (k b)

Definition 1.4

The length of a vector , denoted is defined as

(1.5)           

In Mathematica, we define the length of a vector using the dot product:

lengthn[v_]:=N[Sqrt[v.v]]

lengthn[a]

lengthn[b]

lengthn[c]

Throughout this book, we will be concerned only with vectors in and vectors in .

Theorem 1.5

If a and b are any two vectors in either or , then the dot product of a and b is defined as

(1.6)            .

where θ is the angle between a and b.

The angle between two vectors can be obtained from equation (1.6)

(1.7) ,

where .

So, we can compute the angle between two vectors a and b in Mathematica using

θ=ArcCos[a.b/(lengthn[a] lengthn[b])]

Notice that the angle is in radians.

To convert this value to degrees, type

θ*180/π

The notion of the angle between two vectors gives rise to the idea of the projection of b onto a, written as

(1.8)

The magnitude of is defined as

,

where θ is the angle between a and b.

The direction of is either the same as that of a or opposite to a depending on the value of cos θ.

Thus, the magnitude is

(1.9)

.

Hence, the projection of b onto a is

(1.10)         

.

1.5 The Cross Product of Vectors in

Definition 1.6

Let a and b be two vectors in . The cross product of a and b is denoted by . The length of is defined as

(1.11)           

,

where θ is the angle between a and b.

Note that the cross product of vectors only work on three-dimensional vectors.

There are two ways to compute the cross product of vectors in Mathematica:

1. Using the Cross command

2. Using the sign

(Notes: to generate the sign, press the Esc key, then type the word cross and followed by another Esc key.)

a={1,2,-1};

b={3,2,4};

Cross[a,b]

a b

Properties of the cross products.

If and , then

1. a b= - b a (anti-commutativity)

2. a (b+c)=a b+a c (distributivity)

3. (a+b) c=a c+b c (distributivity)

4. k (a b)=(k a) b= a (k b)

a={a1,a2,a3};

b={b1,b2,b3};

a b

From the result above we see that

Written in vector form

The cross products can also be written in determinant form as

(1.12)         

Using Mathematica, it is straightforward to prove the properties of the cross products.

1.5.1 The Area of a Parallelogram

The area of a parallelogram spanned by two vectors a and b is given by

(1.13)         

a={-1,-2,0};

b={-3,1,0};

(* The area of parallelogram spanned by a and b *)

Sqrt[(Cross[a,b]).( Cross[a,b])]

lengthn[Cross[a,b]]

Alternatively, we can use the Norm command to compute the norm of a b, that is .

Norm[Cross[a,b]]

1.5.2 The Volume of a Parallelepiped

The combination of dot product and cross product can be used to find the volume of a parallelepiped.

Let a parallelepiped be spanned by the vectors a, b, and c. The volume of the parallelepiped is defined as

(1.14)         

The identity is known as triple scalar product.

a={3,-1,0};

b={-2,0,1};

c={1,-2,4};

(* The volume of the parallelepiped *)

Abs[a.Cross[b,c]]

Abs[b.Cross[c,a]]

Abs[c.Cross[a,b]]

Note that we have used the Abs command to compute the absolute value of the volume.

a={a1,a2,a3};

b={b1,b2,b3};

c={c1,c2,c3};

We can use Mathematica to verify the following identity

c.(Cross[a,b])==Det[{c,a,b}]//Simplify

Example 1.4. What is the volume of the parallelepiped with vertices (3,0,-1), (4,2,-1), (-1,1,0), (3,1,5), (0,3,0), (4,3,5), (-1,2,6), and (0,4,6)?

Solution:

Write the vertices as

a=(3,0,-1),         b=(4,2,-1), c=(-1,1,0),

d=(3,1,5),           e=(0,3,0), f=(4,3,5),

g=(-1,2,6),         h=(0,4,6).

a={3,0,-1};

b={4,2,-1};

c={-1,1,0};

d={3,1,5};

e={0,3,0};

f={4,3,5};

g={-1,2,6};

h={0,4,6};

Graphics3D[{PointSize[0.03],Point[{a,b,c,d,e,f,g,h}],Text[Style[a,Blue,Bold,16],a+0.2],Text[Style[b,Blue,Bold,16],b+0.2],Text[Style[c,Blue,Bold,16],c+0.2],Text[Style[d,Blue,Bold,16],d+0.2],Text[Style[e,Blue,Bold,16],e+0.2],Text[Style[f,Blue,Bold,16],f+0.2],Text[Style[g,Blue,Bold,16],g+0.2],Text[Style[h,Blue,Bold,16],h+0.2],{Line[{{a,b},{a,c},{a,d}}]},{Dashed,Line[{{e,c},{e,b},{h,g},{h,f},{d,g},{d,f},{c,g},{e,h},{b,f}}]}},Axes->True,ImageSize->300]

Figure 1.9. The parallelepiped with vertices (3,0,-1), (4,2,-1), (-1,1,0), (3,1,5), (0,3,0), (4,3,5), (-1,2,6), and (0,4,6).

We can find the volume of the parallelepiped by using equation (1.14):

.

Abs[Cross[b-a, c-a].(d-a)]

1.6 Equations for Planes

1.6.1 Coordinate Equation of Planes

Definition 1.7

Let be a point that lies in a plane Π and n=A i+B j+C k a normal vector to Π.  Let P(x,y,z) be any point in the plane. Then Π is defined as the vector equation