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Lecture Notes

Vector Analysis

MATH 332

Ivan Avramidi

New Mexico Institute of Mining and Technology

Socorro, NM 87801

May 19, 2004

Author: Ivan Avramidi; File: vecanal4.tex; Date: July 1, 2005; Time: 13:

Contents

1 Linear Algebra 1

1.1 Vectors in R

n

and Matrix Algebra................... 1

1.1.1 Vectors............................. 1

1.1.2 Matrices............................. 3

1.1.3 Determinant........................... 8

1.1.4 Exercises............................ 9

1.2 Vector Spaces.............................. 11

1.2.1 Exercises............................ 13

1.3 Inner Product and Norm........................ 14

1.3.1 Exercises............................ 15

1.4 Linear Operators............................ 17

1.4.1 Exercises............................ 24

2 Vector and Tensor Algebra 27

2.1 Metric Tensor.............................. 27

2.2 Dual Space and Covectors....................... 29

2.2.1 Einstein Summation Convention................ 31

2.3 General Definition of a Tensor..................... 34

2.3.1 Orientation, Pseudotensors and Volume............ 37

2.4 Operators and Tensors......................... 41

2.5 Vector Algebra in R

3

.......................... 44

3 Geometry 49

3.1 Geometry of Euclidean Space..................... 49

3.2 Basic Topology of R

n

.......................... 53

3.3 Curvilinear Coordinate Systems.................... 54

3.3.1 Change of Coordinates..................... 56

3.3.2 Examples............................ 57

3.4 Vector Functions of a Single Variable................. 59

3.5 Geometry of Curves........................... 61

3.6 Geometry of Surfaces......................... 65

I

• 4 Vector Analysis
  • 4.1 Vector Functions of Several Variables
  • 4.2 Directional Derivative and the Gradient
  • 4.3 Exterior Derivative
  • 4.4 Divergence
  • 4.5 Curl
  • 4.6 Laplacian
  • 4.7 Differential Vector Identities
• 5 Integration
  • 5.1 Line Integrals
  • 5.2 Surface Integrals
  • 5.3 Volume Integrals
  • 5.4 Fundamental Integral Theorems
    • 5.4.1 Fundamental Theorem of Line Integrals
    • 5.4.2 Green’s Theorem
    • 5.4.3 Stokes’s Theorem
    • 5.4.4 Gauss’s Theorem
    • 5.4.5 General Stokes’s Theorem
• 6 Potential Theory
  • 6.1 Simply Connected Domains
  • 6.2 Conservative Vector Fields
    • 6.2.1 Scalar Potential
  • 6.3 Irrotational Vector Fields
  • 6.4 Solenoidal Vector Fields
    • 6.4.1 Vector Potential
  • 6.5 Laplace Equation
    • 6.5.1 Harmonic Functions
  • 6.6 Poisson Equation
    • 6.6.1 Dirac Delta Function
    • 6.6.2 Point Sources
    • 6.6.3 Dirichlet Problem
    • 6.6.4 Neumann Problem
    • 6.6.5 Green’s Functions
  • 6.7 Fundamental Theorem of Vector Analysis
• 7 Basic Concepts of Di ff erential Geometry
  • 7.1 Manifolds
  • 7.2 Differential Forms
    • 7.2.1 Exterior Product
    • 7.2.2 Exterior Derivative
  • 7.3 Integration of Differential Forms
  • 7.4 General Stokes’s Theorem
  • 7.5 Tensors in General Curvilinear Coordinate Systems
    • 7.5.1 Covariant Derivative CONTENTS III
• 8 Applications
  • 8.1 Mechanics
    • 8.1.1 Inertia Tensor
    • 8.1.2 Angular Momentum Tensor
  • 8.2 Elasticity
    • 8.2.1 Strain Tensor
    • 8.2.2 Stress Tensor
  • 8.3 Fluid Dynamics
    • 8.3.1 Continuity Equation
    • 8.3.2 Tensor of Momentum Flux Density
    • 8.3.3 Euler’s Equations
    • 8.3.4 Rate of Deformation Tensor
    • 8.3.5 Navier-Stokes Equations
  • 8.4 Heat and Diffusion Equations
  • 8.5 Electrodynamics
    • 8.5.1 Tensor of Electromagnetic Field
    • 8.5.2 Maxwell Equations
    • 8.5.3 Scalar and Vector Potentials
    • 8.5.4 Wave Equations
    • 8.5.5 D’Alambert Operator
    • 8.5.6 Energy-Momentum Tensor
  • 8.6 Basic Concepts of Special and General Relativity
• Bibliography
• Notation
• Index

IV CONTENTS

2 CHAPTER 1. LINEAR ALGEBRA

• The addition of vectors is defined by

u + v =

u 1

• v 1

u 2

• v 2

u n

• v n

and

u + v = ( u 1

• v 1 ,... , u n
• v n

• Notice that one cannot add a column-vector and a row-vector!
• The multiplication of vectors by a real constant, called a scalar , is defined by

a v =

av 1

av 2

av n

, a v = ( av 1

,... , av n

• The vectors that have only zero elements are called zero vectors , that is

T

= (0,... , 0).

• The set of column-vectors

e 1

, e 2

, · · · , e n

and the set of row-vectors

e

T

1

= (1, 0 ,... , 0), e

T

2

= (0, 1 ,... , 0), e

T

n

are called the standard (or canonical) bases in R

n

.

• There is a natural product of column-vectors and row-vectors that assigns to a

row-vector and a column-vector a real number

〈 u

T

, v 〉 = ( u 1 , u 2 ,... , u n

v 1

v 2

v n

n ∑

i = 1

u i v i = u 1 v 1

• u 2 v 2
• · · · + u n v n

This is the simplest instance of a more general multiplication rule for matrices

which can be summarized by saying that one multiplies row by column.

1.1. VECTORS IN R

N

AND MATRIX ALGEBRA 3

• The product of two column-vectors and the product of two row-vectors, called

the inner product (or the scalar product ), is defined then by

( u , v ) = ( u

T

, v

T

) = 〈 u

T

, v 〉 =

n ∑

i = 1

u i

v i

= u 1

v 1

• · · · + u n

v n

• Finally, we define the norm (or the length ) of both column-vectors and row-

vectors is defined by

|| v || = || v

T

|| =

〈 v

T , v 〉 =

n ∑

i = 1

v

2

i

1 / 2

v

2

1

• · · · + v

2

n

1.1.2 Matrices

• A set of n

2

real numbers A i j , i , j = 1 ,... , n , arranged in an array that has n

columns and n rows

A =

A

11

A

12

· · · A

1 n

A

21

A

22

· · · A

2 n

A

n 1

A

n 2

· · · A

nn

is called a square n × n real matrix.

• The set of all real square n × n matrices is denoted by Mat( n , R).
• The number A i j

(also called an entry of the matrix) appears in the i -th row and

the j -th column of the matrix A

A =

A

11

A

12

· · · A

1 j

· · · A

1 n

A

21

A

22

· · · A

2 j

· · · A

2 n

A

i 1

A

i 2

· · · A

i j

· · · A

in

A

n 1

A

n 2

· · · A

n j

· · · A

nn

• Remark. Notice that the first index indicates the row and the second index

indicates the column of the matrix.

• The matrix whose all entries are equal to zero is called the zero matrix.
• The addition of matrices is defined by

A + B =

A

11

+ B

11

A

12

+ B

12

· · · A

1 n

+ B

1 n

A

21

+ B

21

A

22

+ B

22

· · · A

2 n

+ B

2 n

A

n 1

+ B

n 1

A

n 2

+ B

n 2

· · · A

nn

+ B

nn

1.1. VECTORS IN R

N

AND MATRIX ALGEBRA 5

• A matrix A of the form

A =

where ∗ represents nonzero entries is called an upper triangular matrix. Its

lower triangular part is zero, that is,

A

i j = 0 if i < j.

• A matrix A of the form

A =

whose upper triangular part is zero, that is,

A

i j

= 0 if i > j ,

is called a lower triangular matrix.

• The transpose of a matrix A whose i j -th entry is A i j

is the matrix A

T whose

i j -th entry is A ji

. That is, A

T obtained from A by switching the roles of rows and

columns of A :

A

T

A

11

A

21

· · · A

j 1

· · · A

n 1

A

12

A

22

· · · A

j 2

· · · A

n 2

A

1 i

A

2 i

· · · A

ji

· · · A

ni

A

11

A

2 n

· · · A

jn

· · · A

nn

or

( A

T

) i j

= A

ji

• A matrix A is called symmetric if

A

T

= A

and anti-symmetric if

A

T

= − A.

• The number of independent entries of an anti-symmetric matrix is n ( n − 1)/2.
• The number of independent entries of a symmetric matrix is n ( n + 1)/2.

6 CHAPTER 1. LINEAR ALGEBRA

• Every matrix A can be uniquely decomposed as the sum of its diagonal part A D

the lower triangular part A L and the upper triangular part A U

A = A

D

+ A

L

+ A

U

• For an anti-symmetric matrix

A

T

U

= − A

L and A D

• For a symmetric matrix

A

T

U

= A

L

• Every matrix A can be uniquely decomposed as the sum of its symmetric part A S

and its anti-symmetric part A A

A = A

S

+ A

A

where

A

S

( A + A

T

) , A A

( A − A

T

).

• The product of matrices is defined as follows. The i j -th entry of the product

C = AB of two matrices A and B is

C

i j

n ∑

k = 1

A

ik

B

k j

= A

i 1

B

1 j

+ A

i 2

B

2 j

+ · · · + A

in

B

n j

This is again a multiplication of the “ i -th row of the matrix A by the j -th column

of the matrix B ”.

• Theorem 1.1.1 The product of matrices is associative , that is, for any matrices

A, B, C

( AB ) C = A ( BC ).

• Theorem 1.1.2 For any two matrices A and B

( AB )

T

= B

T

A

T

.

• A matrix A is called invertible if there is another matrix A

− 1 such that

AA

− 1

= A

− 1

A = I.

The matrix A

− 1 is called the inverse of A.

• Theorem 1.1.3 For any two invertible matrices A and B

( AB )

− 1

= B

− 1

A

− 1

,

and

( A

− 1

)

T

= ( A

T

)

− 1

.

8 CHAPTER 1. LINEAR ALGEBRA

1.1.3 Determinant

• Consider the set Z n = { 1 , 2 ,... , n } of the first n integers. A permutation ϕ of the

set { 1 , 2 ,... , n } is an ordered n -tuple (ϕ(1),... , ϕ( n )) of these numbers.

• That is, a permutation is a bijective (one-to-one and onto) function

ϕ : Z n

→ Z

n

that assigns to each number i from the set Z n = { 1 ,... , n } another number ϕ( i )

from this set.

• An elementary permutation is a permutation that exchanges the order of only

two numbers.

• Every permutation can be realized as a product (or a composition) of elemen-

tary permutations. A permutation that can be realized by an even number of

elementary permutations is called an even permutation. A permutation that

can be realized by an odd number of elementary permutations is called an odd

permutation.

• Proposition 1.1.1 The parity of a permutation does not depend on the repre-

sentation of a permutation by a product of the elementary ones.

• That is, each representation of an even permutation has even number of elemen-

tary permutations, and similarly for odd permutations.

• The sign of a permutation ϕ, denoted by sign(ϕ) (or simply (−1)

ϕ

), is defined

by

sign(ϕ) = (−1)

ϕ

=

• 1 , if ϕ is even,

− 1 , if ϕ is odd

• The set of all permutations of n numbers is denoted by S n

• Theorem 1.1.5 The cardinality of this set, that is, the number of di ff erent per-

mutations, is

| S

n

| = n!.

• The determinant is a map det : Mat( n , R) → R that assigns to each matrix

A = ( A

i j ) a real number det A defined by

det A =

ϕ∈ S n

sign (ϕ) A 1 ϕ(1)

· · · A

n ϕ( n )

where the summation goes over all n! permutations.

• The most important properties of the determinant are listed below:

Theorem 1.1.6 1. The determinant of the product of matrices is equal to the

product of the determinants:

det( AB ) = det A det B.

1.1. VECTORS IN R

N

AND MATRIX ALGEBRA 9

2. The determinants of a matrix A and of its transpose A

T

are equal:

det A = det A

T

.

3. The determinant of the inverse A

− 1

of an invertible matrix A is equal to the

inverse of the determinant of A:

det A

− 1

= (det A )

− 1

4. A matrix is invertible if and only if its determinant is non-zero.

• The set of real invertible matrices (with non-zero determinant) is denoted by

GL ( n , R). The set of matrices with positive determinant is denoted by GL

( n , R).

• A matrix with unit determinant is called unimodular.
• The set of real matrices with unit determinant is denoted by S L ( n , R).
• The set of real orthogonal matrices is denoted by O ( n ).
• Theorem 1.1.7 The determinant of an orthogonal matrix is equal to either 1 or

• An orthogonal matrix with unit determinant (a unimodular orthogonal matrix) is

called a proper orthogonal matrix or just a rotation.

• The set of real orthogonal matrices with unit determinant is denoted by S O ( n ).
• A set G of invertible matrices forms a group if it is closed under taking inverse

and matrix multiplication, that is, if the inverse A

− 1 of any matrix A in G belongs

to the set G and the product AB of any two matrices A and B in G belongs to G.

1.1.4 Exercises

1. Show that the product of invertible matrices is an invertible matrix.
2. Show that the product of matrices with positive determinant is a matrix with positive

determinant.

3. Show that the inverse of a matrix with positive determinant is a matrix with positive

determinant.

4. Show that GL ( n , R) forms a group (called the general linear group ).
5. Show that GL
   • ( n , R) is a group (called the proper general linear group ).
6. Show that the inverse of a matrix with negative determinant is a matrix with negative

determinant.

7. Show that: a) the product of an even number of matrices with negative determinant is a

matrix with positive determinant, b) the product of odd matrices with negative determinant

is a matrix with negative determinant.

8. Show that the product of matrices with unit determinant is a matrix with unit determinant.
9. Show that the inverse of a matrix with unit determinant is a matrix with unit determinant.

1.2. VECTOR SPACES 11

1.2 Vector Spaces

• A real vector space consists of a set E , whose elements are called vectors , and

the set of real numbers R, whose elements are called scalars. There are two

operations on a vector space:

1. Vector addition , + : E × E → E , that assigns to two vectors u , v ∈ E

another vector u + v , and

2. Multiplication by scalars , · : R × E → E , that assigns to a vector v ∈ E

and a scalar a ∈ R a new vector a v ∈ E.

The vector addition is an associative commutative operation with an additive

identity. It satisfies the following conditions:

1. u + v = v + u , ∀ u , v , ∈ E
2. ( u + v ) + w = u + ( v + w ), ∀ u , v , w ∈ E
3. There is a vector 0 ∈ E , called the zero vector , such that for any v ∈ E

there holds v + 0 = v.

4. For any vector v ∈ E , there is a vector (− v ) ∈ E , called the opposite of v ,

such that v + (− v ) = 0.

The multiplication by scalars satisfies the following conditions:

1. a ( b v ) = ( ab ) v , ∀ v ∈ E , ∀ a , b R,
2. ( a + b ) v = a v + b v , ∀ v ∈ E , ∀ a , b R,
3. a ( u + v ) = a u + a v , ∀ u , v ∈ E , ∀ a R,
4. 1 v = v ∀ v ∈ E.

• The zero vector is unique.
• For any u , v ∈ E there is a unique vector denoted by w = v − u , called the

difference of v and u , such that u + w = v.

• For any v ∈ E ,

0 v = 0 , and (−1) v = − v.

• Let E be a real vector space and A = { e 1

,... , e k

} be a finite collection of vectors

from E. A linear combination of these vectors is a vector

a 1 e 1

• · · · + a k e k

where { a 1

,... , a n

} are scalars.

• A finite collection of vectors A = { e 1 ,... , e k } is linearly independent if

a 1

e 1

• · · · + a k

e k

implies a 1 = · · · = a k

12 CHAPTER 1. LINEAR ALGEBRA

• A collection A of vectors is linearly dependent if it is not linearly independent.
• Two non-zero vectors u and v which are linearly dependent are also called par-

allel , denoted by u || v.

• A collection A of vectors is linearly independent if no vector of A is a linear

combination of a finite number of vectors from A.

• Let A be a subset of a vector space E. The span of A, denoted by span A, is the

subset of E consisting of all finite linear combinations of vectors from A, i.e.

span A = { v ∈ E | v = a 1 e 1

• · · · + a k e k , e i ∈ A, a i

∈ R}.

We say that the subset span A is spanned by A.

• Theorem 1.2.1 The span of any subset of a vector space is a vector space.
• A vector subspace of a vector space E is a subset S ⊆ E of E which is itself a

vector space.

• Theorem 1.2.2 A subset S of E is a vector subspace of E if and only if span S =

S.

• Span of A is the smallest subspace of E containing A.
• A collection B of vectors of a vector space E is a basis of E if B is linearly

independent and span B = E.

• A vector space E is finite-dimensional if it has a finite basis.
• Theorem 1.2.3 If the vector space E is finite-dimensional, then the number of

vectors in any basis is the same.

• The dimension of a finite-dimensional real vector space E , denoted by dim E , is

the number of vectors in a basis.

• Theorem 1.2.4 If { e 1 ,... , e n } is a basis in E, then for every vector v ∈ E there

is a unique set of real numbers ( v

i

) = ( v

1

,... , v

n

) such that

v =

n ∑

i = 1

v

i

e i

= v

1

e 1

• · · · + v

n

e n

• The real numbers v

i

, i = 1 ,... , n , are called the components of the vector v

with respect to the basis { e i

• It is customary to denote the components of vectors by superscripts , which

should not be confused with powers of real numbers

v

2

, ( v )

2

= vv ,... , v

n

, ( v )

n

.

14 CHAPTER 1. LINEAR ALGEBRA

1.3 Inner Product and Norm

• A real vector space E is called an inner product space if there is a function

(·, ·) : E × E → R, called the inner product , that assigns to every two vectors u

and v a real number ( u , v ) and satisfies the conditions: ∀ u , v , w ∈ E , ∀ a ∈ R:

1. ( v , v ) ≥ 0
2. ( v , v ) = 0 if and only if v = 0
3. ( u , v ) = ( v , u )
4. ( u + v , w ) = ( u , w ) + ( v , w )
5. ( a u , v ) = ( u , a v ) = a ( u , v )

A finite-dimensional inner product space is called a Euclidean space.

• The inner product is often called the dot product , or the scalar product , and is

denoted by

( u , v ) = u · v.

• All spaces considered below are Euclidean spaces. Henceforth, E will denote an

n -dimensional Euclidean space if not specified otherwise.

• The Euclidean norm is a function || · || : E → R that assigns to every vector

v ∈ E a real number || v || defined by

|| v || =

( v , v ).

• The norm of a vector is also called the length.
• A vector with unit norm is called a unit vector.
• Theorem 1.3.1 For any u , v ∈ E there holds

|| u + v ||

2

= || u ||

2

• 2( u , v ) + || v ||

2

.

• Theorem 1.3.2 Cauchy-Schwarz’s Inequality. For any u , v ∈ E there holds

|( u , v )| ≤ || u || || v ||.

The equality

|( u , v )| = || u || || v ||

holds if and only if u and v are parallel.

• Corollary 1.3.1 Triangle Inequality. For any u , v ∈ E there holds

|| u + v || ≤ || u || + || v ||.

1.3. INNER PRODUCT AND NORM 15

• The angle between two non-zero vectors u and v is defined by

cos θ =

( u , v )

|| u || || v ||

, 0 ≤ θ ≤ π.

Then the inner product can be written in the form

( u , v ) = || u || || v || cos θ.

• Two non-zero vectors u , v ∈ E are orthogonal , denoted by u ⊥ v , if

( u , v ) = 0.

• A basis { e 1

,... , e n

} is called orthonormal if each vector of the basis is a unit

vector and any two distinct vectors are orthogonal to each other, that is,

( e i , e j

1 , if i = j

0 , if i , j

• Theorem 1.3.3 Every Euclidean space has an orthonormal basis.
• Let S ⊂ E be a nonempty subset of E. We say that x ∈ E is orthogonal to S ,

denoted by x ⊥ S , if x is orthogonal to every vector of S.

• The set

S

⊥

= { x ∈ E | x ⊥ S }

of all vectors orthogonal to S is called the orthogonal complement of S.

• Theorem 1.3.4 The orthogonal complement of any subset of a Euclidean space

is a vector subspace.

• Two subsets A and B of E are orthogonal , denoted by A ⊥ B , if every vector of

A is orthogonal to every vector of B.

• Let S be a subspace of E and S

⊥ be its orthogonal complement. If every element

of E can be uniquely represented as the sum of an element of S and an element

of S

⊥ , then E is the direct sum of S and S

⊥ , which is denoted by

E = S ⊕ S

⊥

.

• The union of a basis of S and a basis of S

⊥

gives a basis of E.

1.3.1 Exercises

1. Show that the Euclidean norm has the following properties

(a) || v || ≥ 0, ∀ v ∈ E ;

(b) || v || = 0 if and only if v = 0;