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Vector Analysis

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Vector Analysis

and an introduction to

TENSOR ANALYSIS

Second Edition

Seymour Lipschutz, Ph.D.

Mathematics Department, Temple University

Dennis Spellman, Ph.D.

Mathematics Department, Temple University

Murray R. Spiegel, Ph.D.

Former Professor and Chairman, Mathematics Department Rensselaer Polytechnic Institute, Hartford Graduate Center

Schaum’s Outline Series

New York Chicago San Francisco Lisbon London Madrid Mexico City Milan New Delhi San Juan Seoul Singapore Sydney Toronto

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SEYMOUR LIPSCHUTZ is on the faculty of Temple University and formerly taught at the Polytechnic Institute of Brooklyn. He received his Ph.D. from New York University and is one of Schaum’s most prolific authors. In particular he has written, among others, Linear Algebra, Probability, Discrete Mathematics, Set Theory, Finite Mathematics, and General Topology.

DENNIS SPELLMAN is on the faculty of Temple University and he formerly taught at the University of the East in Venezuela. He received his Ph.D. from New York University where he wrote his thesis with Wilhelm Magnus. He is the author of over twenty-five journal articles in pure and applied mathematics.

The late MURRAY R. SPIEGEL received the MS degree in Physics and the Ph.D. degree in Mathematics from Cornell University. He had positions at Harvard University, Columbia University, Oak Ridge, and Rensselaer Polytechnic Institute, and served as a mathematical consultant at several large companies. His last position was Professor and Chairman of Mathematics at the Rensselaer Polytechnic Institute, Hartford Graduate Center. He was interested in most branches of mathematics, especially those that involve applications to physics and engineering problems. He was the author of numerous journal articles and 14 books on various topics in mathematics.

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Contents

CHAPTER 1 VECTORS AND SCALARS 1

1.1 Introduction 1.2 Vector Algebra 1.3 Unit Vectors 1.4 Rectangular Unit Vectors i, j, k 1.5 Linear Dependence and Linear Independence 1.6 Scalar Field 1.7 Vector Field 1.8 Vector Space Rn

CHAPTER 2 THE DOT AND CROSS PRODUCT 21

2.1 Introduction 2.2 Dot or Scalar Product 2.3 Cross Product 2.4 Triple Products 2.5 Reciprocal Sets of Vectors

CHAPTER 3 VECTOR DIFFERENTIATION 44

3.1 Introduction 3.2 Ordinary Derivatives of Vector-Valued Functions 3.3 Continuity and Differentiability 3.4 Partial Derivative of Vectors 3.5 Differential Geometry

CHAPTER 4 GRADIENT, DIVERGENCE, CURL 69

4.1 Introduction 4.2 Gradient 4.3 Divergence 4.4 Curl 4.5 Formulas Involving 7 4.6 Invariance

CHAPTER 5 VECTOR INTEGRATION 97

5.1 Introduction 5.2 Ordinary Integrals of Vector Valued Functions 5.3 Line Integrals 5.4 Surface Integrals 5.5 Volume Integrals

CHAPTER 6 DIVERGENCE THEOREM, STOKES’ THEOREM,

AND RELATED INTEGRAL THEOREMS 126

6.1 Introduction 6.2 Main Theorems 6.3 Related Integral Theorems

CHAPTER 7 CURVILINEAR COORDINATES 157

7.1 Introduction 7.2 Transformation of Coordinates 7.3 Orthogonal Curvi- linear Coordinates 7.4 Unit Vectors in Curvilinear Systems 7.5 Arc Length and Volume Elements 7.6 Gradient, Divergence, Curl 7.7 Special Orthog- onal Coordinate Systems

vii

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Vector Analysis

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numbers. Thus we can denote them, as usual, by ordinary letters. Also, the real numbers 0 and 1 are part of our set of scalars.

1.2 Vector Algebra

There are two basic operations with vectors: (a) Vector Addition; (b) Scalar Multiplication.

(a) Vector Addition

Consider vectors A and B, pictured in Fig. 1-2(a). The sum or resultant of A and B, is a vector C formed by placing the initial point of B on the terminal point of A and then joining the initial point of A to the terminal point of B, pictured in Fig. 1-2(b). The sum C is written C ¼ A þ B. This definition here is equivalent to the Parallelogram Law for vector addition, pictured in Fig. 1-2(c).

A^ B

C = A + B

(b)

B

A

(a)

B

A C = A + B

(c) Fig. 1-

Extensions to sums of more than two vectors are immediate. Consider, for example, vectors A, B, C, D in Fig. 1-3(a). Then Fig. 1-3(b) shows how to obtain the sum or resultant E of the vectors A, B, C, D, that is, by connecting the end of each vector to the beginning of the next vector.

A

C

D

B

(a)

A

B

C

D

(b)

E = A + B + C + D

Fig. 1-

The difference of vectors A and B, denoted by A  B, is that vector C, which added to B, gives A. Equivalently, A  B may be defined as A þ (B). If A ¼ B, then A  B is defined as the null or zero vector; it is represented by the symbol 0 or 0. It has zero magnitude and its direction is undefined. A vector that is not null is a proper vector. All vectors will be assumed to be proper unless otherwise stated.

(b) Scalar Multiplication

Multiplication of a vector A by a scalar m produces a vector mA with magnitude jmj times the magnitude of A and the direction of mA is in the same or opposite of A according as m is positive or negative. If m ¼ 0, then mA ¼ 0 , the null vector.

2 CHAPTER 1 Vectors and Scalars

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Laws of Vector Algebra

The following theorem applies.

THEOREM 1.1: Suppose A, B, C are vectors and m and n are scalars. Then the following laws hold:

[A 1 ] (A þ B) þ C ¼ (A þ B) þ C Associative Law for Addition [A 2 ] There exists a zero vector 0 such that, for every vector A, A þ 0 ¼ 0 þ A ¼ A Existence of Zero Element

[A 3 ] For every vector A, there exists a vector A such that A þ (A) ¼ (A) þ A ¼ 0 Existence of Negatives

[A 4 ] A þ B ¼ B þ A Commutative Law for Addition [M 1 ] m(A þ B) ¼ mA þ mB Distributive Law [M 2 ] (m þ n)A ¼ mA þ nA Distributive Law [M 3 ] m(nA) ¼ (mn)A Associative Law [M 4 ] 1(A) ¼ A Unit Multiplication

The above eight laws are the axioms that define an abstract structure called a vector space. The above laws split into two sets, as indicated by their labels. The first four laws refer to vector addition. One can then prove the following properties of vector addition.

(a) Any sum A 1 þ A 2 þ    þ An of vectors requires no parentheses and does not depend on the order of the summands. (b) The zero vector 0 is unique and the negative A of a vector A is unique. (c) (Cancellation Law) If A þ C ¼ B þ C, then A ¼ B. The remaining four laws refer to scalar multiplication. Using these additional laws, we can prove the following properties.

PROPOSITION 1.2: (a) For any scalar m and zero vector 0 , we have m 0 ¼ 0. (b) For any vector A and scalar 0, we have 0A ¼ 0. (c) If mA ¼ 0 , then m ¼ 0 or A ¼ 0. (d) For any vector A and scalar m, we have (m)A ¼ m(A) ¼ (mA).

1.3 Unit Vectors

Unit vectors are vectors having unit length. Suppose A is any vector with length jAj. 0. Then A=jAj is a unit vector, denoted by a, which has the same direction as A. Also, any vector A may be represented by a unit vector a in the direction of A multiplied by the magnitude of A. That is, A ¼ jAja.

EXAMPLE 1.1 Suppose jAj ¼ 3. Then a ¼ jAj=3 is a unit vector in the direction of A. Also, A ¼ 3 a.

1.4 Rectangular Unit Vectors i, j, k

An important set of unit vectors, denoted by i, j, and k, are those having the directions, respectively, of the positive x, y, and z axes of a three-dimensional rectangular coordinate system. [See Fig. 1-4(a).] The coordinate system shown in Fig. 1-4(a), which we use unless otherwise stated, is called a right- handed coordinate system. The system is characterized by the following property. If we curl the fingers of the right hand in the direction of a 90 8 rotation from the positive x-axis to the positive y-axis, then the thumb will point in the direction of the positive z-axis.

CHAPTER 1 Vectors and Scalars 3

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1.5 Linear Dependence and Linear Independence

Suppose we are given vectors A 1 , A 2 ,... , An and scalars a 1 , a 2 ,... , an. We can multiply the vectors by the corresponding scalars and then add the corresponding scalar products to form the vector

B ¼ a 1 A 1 þ a 2 A 2 þ    þ anAn

Such a vector B is called a linear combination of the vectors A 1 , A 2 ,... , An. The following definition applies.

DEFINITION Vectors A 1 , A 2 ,... , An are linearly dependent if there exist scalars a 1 , a 2 ,... , an, not all zero, such that a 1 A 1 þ a 2 A 2 þ    þ anAn ¼ 0 Otherwise, the vectors are linearly independent.

The above definition may be restated as follows. Consider the vector equation x 1 A 1 þ x 2 A 2 þ    þ xnAn ¼ 0

where x 1 , x 2 ,... , xn are unknown scalars. This equation always has the zero solution x 1 ¼ 0, x 2 ¼ 0,... , xn ¼ 0. If this is the only solution, the vectors are linearly independent. If there is a solution with some xj = 0, then the vectors are linearly dependent. Suppose A is not the null vector. Then A, by itself, is linearly independent, since mA ¼ 0 and A = 0, implies m ¼ 0

The following proposition applies.

PROPOSITION 1.4: Two or more vectors are linearly dependent if and only if one of them is a linear combination of the others.

COROLLARY 1.5: Vectors A and B are linearly dependent if and only if one is a multiple of the other.

EXAMPLE 1. (a) The unit vectors i, j, k are linearly independent since neither of them is a linear combination of the other two. (b) Suppose aA þ bB þ cC ¼ a^0 A þ b^0 B þ c^0 C where A, B, C are linearly independent. Then a ¼ a^0 , b ¼ b^0 , c ¼ c^0.

1.6 Scalar Field

Suppose that to each point (x, y, z) of a region D in space, there corresponds a number (scalar) f(x, y, z).

Then f is called a scalar function of position, and we say that a scalar field f has been defined on D.

EXAMPLE 1. (a) The temperature at any point within or on the Earth’s surface at a certain time defines a scalar field. (b) The function f(x, y, z) ¼ x^3 y  z^2 defines a scalar field. Consider the point P(2, 3, 1). Then f(P) ¼ 8(3)  1 ¼ 23.

A scalar field f, which is independent of time, is called a stationary or steady-state scalar field.

1.7 Vector Field

Suppose to each point (x, y, z) of a region D in space there corresponds a vector V(x, y, z). Then V is called a vector function of position, and we say that a vector field V has been defined on D.

CHAPTER 1 Vectors and Scalars 5

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EXAMPLE 1. (a) Suppose the velocity at any point within a moving fluid is known at a certain time. Then a vector field is defined. (b) The function V(x, y, z) ¼ xy^2 i  2 yz^3 j þ x^2 zk defines a vector field. Consider the point P(2, 3, 1). Then V(P) ¼ 18 i  6 j þ 4 k. A vector field V which is independent of time is called a stationary or steady-state vector field.

1.8 Vector Space Rn

Let V ¼ Rn^ where Rn^ consists of all n-element sequences u ¼ (a 1 , a 2 ,... , an) of real numbers called the components of u. The term vector is used for the elements of V and we denote them using the letters u, v, and w, with or without a subscript. The real numbers we call scalars and we denote them using letters other than u, v, or w. We define two operations on V ¼ Rn:

(a) Vector Addition

Given vectors u ¼ (a 1 , a 2 ,... , an) and v ¼ (b 1 , b 2 ,... , bn) in V, we define the vector sum u þ v by

u þ v ¼ (a 1 þ b 1 , a 2 þ b 2 ,... , an þ bn)

That is, we add corresponding components of the vectors.

(b) Scalar Multiplication

Given a vector u ¼ (a 1 , a 2 ,... , an) and a scalar k in R, we define the scalar product ku by

ku ¼ (ka 1 , ka 2 ,... , kan) That is, we multiply each component of u by the scalar k.

PROPOSITION 1.6: V ¼ Rn^ satisfies the eight axioms of a vector space listed in Theorem 1.1.

SOLVED PROBLEMS

1.1. State which of the following are scalars and which are vectors: (a) specific heat, (b) momentum, (c) distance, (d) speed, (e) magnetic field intensity

Solution (a) scalar, (b) vector, (c) scalar, (d) scalar, (e) vector

1.2. Represent graphically: (a) a force of 10 lb in a direction 30 8 north of east, (b) a force of 15 lb in a direction 30 8 east of north.

Solution Choosing the unit of magnitude shown, the required vectors are as indicated in Fig. 1-5.

Unit = 5 lb

N

S (a)

30°

10 lb

W E W

N

S (b)

30° 15 lb

E

Fig. 1-

6 CHAPTER 1 Vectors and Scalars

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1.5. Show that addition of vectors is commutative, that is, A þ B ¼ B þ A. (Theorem 1.1 [A 4 ].)

Solution As indicated by Fig. 1-8, OP þ PQ ¼ OQ or A þ B ¼ C and OR þ RQ ¼ OQ or B þ A ¼ C Thus A þ B ¼ B þ A.

O

A

P

R

A

C^ =

B^ +

A C^ =

A^ +^ B

B

B

Q

O

A

P

R

C

( B

• C ( A ) +^ B

)

D

B Q

Fig. 1-8 Fig. 1-

1.6. Show that addition of vectors is associative, that is, A þ (B þ C) ¼ (A þ B) þ C. (Theorem 1.1 [A 1 ].)

Solution As indicated by Fig. 1-9,

OP þ PQ ¼ OQ ¼ (A þ B) and PQ þ QR ¼ PR ¼ (B þ C) OP þ PR ¼ OR ¼ D or A þ (B þ C) ¼ D and OQ þ QR ¼ OR ¼ D or (A þ B) þ C ¼ D Then A þ (B þ C) ¼ (A þ B) þ C.

1.7. Forces F 1 , F 2 ,... , F 6 act on an object P as shown in Fig. 1-10(a). Find the force that is needed to prevent P from moving.

Solution Since the order of addition of vectors is immaterial, we may start with any vector, say F 1. To F 1 add F 2 , then F 3 , and so on as pictured in Fig. 1-10(b). The vector drawn from the initial point of F 1 to the terminal point of F 6 is the resultant R, that is, R ¼ F 1 þ F 2 þ    þ F 6. The force needed to prevent P from moving is R, sometimes called the equilibrant.

F 5

F 5

F (^1) F 1

F 2

F 2 F 3 F 3

F 4

F 4

P P

Resultant =

R

(a) (^) (b)

F 6 F^6

Fig. 1-

8 CHAPTER 1 Vectors and Scalars

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1.8. Given vectors A, B, and C in Fig. 1-11(a), construct A  B þ 2 C.

Solution Beginning with A, we add B and then add 2C as in Fig. 1-11(b). The resultant is A  B þ 2 C.

A

B

C

(a)

A

2 C

A^ –^ B^ + 2

C^ =^ A^ + (–

B ) + 2 C

–B

(b) Fig. 1-

1.9. Given two non-collinear vectors a and b, as in Fig. 1-12. Find an expression for any vector r lying in the plane determined by a and b.

Solution Non-collinear vectors are vectors that are not parallel to the same line. Hence, when their initial points coincide, they determine a plane. Let r be any vector lying in the plane of a and b and having its initial point coincident with the initial points of a and b at O. From the terminal point R of r, construct lines parallel to the vectors a and b and complete the parallelogram ODRC by extension of the lines of action of a and b if necessary. From Fig. 1-12,

OD ¼ x(OA) ¼ xa, where x is a scalar OC ¼ y(OB) ¼ yb, where y is a scalar:

But by the parallelogram law of vector addition

OR ¼ OD þ OC or r ¼ xa þ yb

which is the required expression. The vectors xa and yb are called component vectors of r in the directions a and b, respectively. The scalars x and y may be positive or negative depending on the relative orientations of the vectors. From the manner of construction, it is clear that x and y are unique for a given a, b, and r. The vectors a and b are called base vectors in a plane.

D

A

O C B

R

a r

b

U

a

c

r

T

S

P b

V

A

C

O

R

B

Q

Fig. 1-12 Fig. 1-

CHAPTER 1 Vectors and Scalars 9

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