Análise Matemática: Série Geométrica, Convergência, Funções Trigonométricas e Hiperbólicas, Notas de estudo de Engenharia Mecânica

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Contents

Introduction............................................................................................ 1

Bibliography; Physical Constants

1. Series.................................................................................................... 2

Arithmetic and Geometric progressions; Convergence of series: the ratio test;

Convergence of series: the comparison test; Binomial expansion; Taylor and Maclaurin Series;

Power series with real variables; Integer series; Plane wave expansion

2. Vector Algebra......................................................................................... 3

Scalar product; Equation of a line; Equation of a plane; Vector product; Scalar triple product;

Vector triple product; Non-orthogonal basis; Summation convention

3. Matrix Algebra........................................................................................ 5

Unit matrices; Products; Transpose matrices; Inverse matrices; Determinants; 2 × 2 matrices;

Product rules; Orthogonal matrices; Solving sets of linear simultaneous equations; Hermitian matrices;

Eigenvalues and eigenvectors; Commutators; Hermitian algebra; Pauli spin matrices

4. Vector Calculus........................................................................................ 7

Notation; Identities; Grad, Div, Curl and the Laplacian; Transformation of integrals

5. Complex Variables.................................................................................... 9

Complex numbers; De Moivre’s theorem; Power series for complex variables.

6. Trigonometric Formulae............................................................................ 10

Relations between sides and angles of any plane triangle;

Relations between sides and angles of any spherical triangle

7. Hyperbolic Functions............................................................................... 11

Relations of the functions; Inverse functions

8. Limits.................................................................................................. 12

9. Differentiation........................................................................................ 13

10. Integration............................................................................................ 13

Standard forms; Standard substitutions; Integration by parts; Differentiation of an integral;

Dirac δ -‘function’; Reduction formulae

11. Differential Equations............................................................................... 16

Diffusion (conduction) equation; Wave equation; Legendre’s equation; Bessel’s equation;

Laplace’s equation; Spherical harmonics

12. Calculus of Variations............................................................................... 17

13. Functions of Several Variables..................................................................... 18

Taylor series for two variables; Stationary points; Changing variables: the chain rule;

Changing variables in surface and volume integrals – Jacobians

14. Fourier Series and Transforms..................................................................... 19

Fourier series; Fourier series for other ranges; Fourier series for odd and even functions;

Complex form of Fourier series; Discrete Fourier series; Fourier transforms; Convolution theorem;

Parseval’s theorem; Fourier transforms in two dimensions; Fourier transforms in three dimensions

15. Laplace Transforms.................................................................................. 23

16. Numerical Analysis................................................................................. 24

Finding the zeros of equations; Numerical integration of differential equations;

Central difference notation; Approximating to derivatives; Interpolation: Everett’s formula;

Numerical evaluation of definite integrals

17. Treatment of Random Errors....................................................................... 25

Range method; Combination of errors

18. Statistics............................................................................................... 26

Mean and Variance; Probability distributions; Weighted sums of random variables;

Statistics of a data sample x 1 ,... , xn; Regression (least squares fitting)

1. Series

Arithmetic and Geometric progressions

A.P. Sn = a + ( a + d ) + ( a + 2 d ) + · · · + [ a + ( n − 1 ) d ] =

n

[ 2 a + ( n − 1 ) d ]

G.P. Sn = a + ar + ar

2

• · · · + ar

n − 1 = a

1 − r

n

1 − r

S ∞ =

a

1 − r

for | r | < 1

(These results also hold for complex series.)

Convergence of series: the ratio test

Sn = u 1 + u 2 + u 3 + · · · + un converges as n → ∞ if lim n →∞

un + 1

un

Convergence of series: the comparison test

If each term in a series of positive terms is less than the corresponding term in a series known to be convergent,

then the given series is also convergent.

Binomial expansion

( 1 + x )

n = 1 + nx +

n ( n − 1 )

x

2

n ( n − 1 )( n − 2 )

x

3

• · · ·

If n is a positive integer the series terminates and is valid for all x : the term in x

r is

n Cr x

r or

n

r

where

n Cr ≡

n!

r !( n − r )!

is the number of different ways in which an unordered sample of r objects can be selected from a set of

n objects without replacement. When n is not a positive integer, the series does not terminate: the infinite series is

convergent for | x | < 1.

Taylor and Maclaurin Series

If y ( x ) is well-behaved in the vicinity of x = a then it has a Taylor series,

y ( x ) = y ( a + u ) = y ( a ) + u

d y

d x

u

2

d

2 y

d x 2

u

3

d

3 y

d x 3

where u = x − a and the differential coefficients are evaluated at x = a. A Maclaurin series is a Taylor series with

a = 0,

y ( x ) = y ( 0 ) + x

d y

d x

x

2

d

2 y

d x 2

x

3

d

3 y

d x 3

Power series with real variables

e

x = 1 + x +

x 2

x n

n!

• · · · valid for all x

ln( 1 + x ) = x −

x

2

x 3

n + 1 x n

n

• · · · valid for − 1 < x ≤ 1

cos x =

e

i x

• e

−i x

x

2

x

4

x

6

• · · · valid for all values of x

sin x =

e

i x − e

−i x

2i

= x −

x

3

x

5

• · · · valid for all values of x

tan x = x +

x

3

x

5

• · · · valid for −

π

< x <

π

tan

− 1 x = x −

x

3

x

5

− · · · valid for − 1 ≤ x ≤ 1

sin

− 1 x = x +

x

3

x

5

• · · · valid for − 1 < x < 1

Integer series

N

1

n = 1 + 2 + 3 + · · · + N =

N ( N + 1 )

N

1

n

2 = 1

2

• 2

2

• 3

2

• · · · + N

2

N ( N + 1 )( 2 N + 1 )

N

1

n

3 = 1

3

• 2

3

• 3

3

• · · · + N

3 = [ 1 + 2 + 3 + · · · N ]

2

N

2 ( N + 1 ) 2

∞

1

n + 1

n

• · · · = ln 2 [see expansion of ln( 1 + x )]

∞

1

n + 1

2 n − 1

π

[see expansion of tan

− 1 x ]

∞

1

n

2

π

2

N

1

n ( n + 1 )( n + 2 ) = 1.2.3 + 2.3.4 + · · · + N ( N + 1 )( N + 2 ) =

N ( N + 1 )( N + 2 )( N + 3 )

This last result is a special case of the more general formula,

N

1

n ( n + 1 )( n + 2 )... ( n + r ) =

N ( N + 1 )( N + 2 )... ( N + r )( N + r + 1 )

r + 2

Plane wave expansion

exp(i kz ) = exp(i kr cos θ) =

∞

l = 0

( 2 l + 1 )i

l jl ( kr ) Pl (cos θ),

where Pl (cos θ) are Legendre polynomials (see section 11) and jl ( kr ) are spherical Bessel functions, defined by

jl ( ρ) =

π

2 ρ

Jl + 1 / 2 ( ρ), with Jl ( x ) the Bessel function of order l (see section 11).

2. Vector Algebra

If i , j , k are orthonormal vectors and A = Ax i + Ay j + Az k then | A |

2 = A

2 x +^ A

2 y +^ A

2 z. [Orthonormal vectors^ ≡

orthogonal unit vectors.]

Scalar product

A · B = | A | | B | cos θ where θ is the angle between the vectors

= Ax Bx + Ay By + Az Bz = [ Ax Ay Az ]

Bx

By

Bz

Scalar multiplication is commutative: A · B = B · A.

Equation of a line

A point r ≡ ( x , y , z ) lies on a line passing through a point a and parallel to vector b if

r = a + λ b

with λ a real number.

3. Matrix Algebra

Unit matrices

The unit matrix I of order n is a square matrix with all diagonal elements equal to one and all off-diagonal elements

zero, i.e., ( I ) i j = δ i j. If A is a square matrix of order n , then AI = I A = A. Also I = I

− 1 .

I is sometimes written as In if the order needs to be stated explicitly.

Products

If A is a ( n × l ) matrix and B is a ( l × m ) then the product AB is defined by

( AB ) i j =

l

k = 1

Aik Bk j

In general AB 6 = BA.

Transpose matrices

If A is a matrix, then transpose matrix A

T is such that ( A

T ) i j = ( A ) ji.

Inverse matrices

If A is a square matrix with non-zero determinant, then its inverse A

− 1 is such that AA

− 1 = A

− 1 A = I.

( A

− 1 ) i j =

transpose of cofactor of Ai j

| A |

where the cofactor of Ai j is (− 1 )

i + j times the determinant of the matrix A with the j -th row and i -th column deleted.

Determinants

If A is a square matrix then the determinant of A , | A | (≡ det A ) is defined by

| A | =

i , j , k ,...

 i jk ... A 1 i A 2 j A 3 k...

where the number of the suffixes is equal to the order of the matrix.

2 × 2 matrices

If A =

a b

c d

then,

| A | = ad − bc A

T

a c

b d

A

− 1

| A |

d − b

− c a

Product rules

( AB... N )

T = N

T

... B

T A

T

( AB... N )

− 1 = N

− 1

... B

− 1 A

− 1 (if individual inverses exist)

| AB... N | = | A | | B |... | N | (if individual matrices are square)

Orthogonal matrices

An orthogonal matrix Q is a square matrix whose columns qi form a set of orthonormal vectors. For any orthogonal

matrix Q ,

Q

− 1 = Q

T , | Q | = ±1, Q

T is also orthogonal.

Solving sets of linear simultaneous equations

If A is square then A x = b has a unique solution x = A

− 1 b if A

− 1 exists, i.e., if | A | 6 = 0.

If A is square then A x = 0 has a non-trivial solution if and only if | A | = 0.

An over-constrained set of equations A x = b is one in which A has m rows and n columns, where m (the number

of equations) is greater than n (the number of variables). The best solution x (in the sense that it minimizes the

error | A x − b |) is the solution of the n equations A

T A x = A

T b. If the columns of A are orthonormal vectors then

x = A

T b.

Hermitian matrices

The Hermitian conjugate of A is A

= ( A

∗ )

T , where A

∗ is a matrix each of whose components is the complex

conjugate of the corresponding components of A. If A = A

then A is called a Hermitian matrix.

Eigenvalues and eigenvectors

The n eigenvalues λ i and eigenvectors u i of an n × n matrix A are the solutions of the equation A u = λ u. The

eigenvalues are the zeros of the polynomial of degree n , Pn ( λ) = | A − λ I |. If A is Hermitian then the eigenvalues

λ i are real and the eigenvectors ui are mutually orthogonal. | A − λ I | = 0 is called the characteristic equation of the

matrix A.

Tr A = ∑

i

λ i , also | A | = ∏

i

λ i.

If S is a symmetric matrix, Λ is the diagonal matrix whose diagonal elements are the eigenvalues of S , and U is the

matrix whose columns are the normalized eigenvectors of A , then

U

T SU = Λ and S = U Λ U

T .

If x is an approximation to an eigenvector of A then x

T A x /( x

T x ) (Rayleigh’s quotient) is an approximation to the

corresponding eigenvalue.

Commutators

[ A , B ] ≡ AB − BA

[ A , B ] = −[ B , A ]

[ A , B ]

= [ B

, A

]

[ A + B , C ] = [ A , C ] + [ B , C ]

[ AB , C ] = A [ B , C ] + [ A , C ] B

[ A , [ B , C ]] + [ B , [ C , A ]] + [ C , [ A , B ]] = 0

Hermitian algebra

b

= ( b

∗ 1 , b

∗ 2

Matrix form Operator form Bra-ket form

Hermiticity b

∗ · A · c = ( A · b )

∗ · c

Z

ψ

∗ O φ =

Z

( O ψ)

∗ φ 〈 ψ| O | φ〉

Eigenvalues, λ real A u i = λ( i ) u i O ψ i = λ( i ) ψ i O | i 〉 = λ i | i 〉

Orthogonality u i · u j = 0

Z

ψ

∗ i ψ j = 0 〈 i | j 〉 = 0 ( i 6 = j )

Completeness b = ∑

i

u i ( u i · b ) φ = ∑

i

ψ i

(Z

ψ

∗ i φ

i

| i 〉 〈 i | φ〉

Rayleigh–Ritz

Lowest eigenvalue λ 0 ≤

b

∗ · A · b

b

∗ · b

λ 0 ≤

Z

ψ

∗ O ψ

Z

ψ

∗ ψ

〈 ψ| O | ψ〉

〈 ψ| ψ〉

Grad, Div, Curl and the Laplacian

Cartesian Coordinates Cylindrical Coordinates Spherical Coordinates

Conversion to

Cartesian

Coordinates

x = ρ cos ϕ y = ρ sin ϕ z = z

x = r cos ϕ sin θ y = r sin ϕ sin θ

z = r cos θ

Vector A Ax i + Ay j + Az k A ρ ρ̂ + A ϕ ϕ̂ + Az ̂ z Ar ̂ r + A θ θ̂ + A ϕϕ̂

Gradient ∇ φ

∂ φ

∂ x

i +

∂ φ

∂ y

j +

∂ φ

∂ z

k

∂ φ

∂ ρ

ρ +

ρ

∂ φ

∂ ϕ

ϕ +

∂ φ

∂ z

z

∂ φ

∂ r

r +

r

∂ φ

∂ θ

θ +

r sin θ

∂ φ

∂ ϕ

ϕ

Divergence

∇ · A

∂ Ax

∂ x

∂ Ay

∂ y

∂ Az

∂ z

ρ

∂( ρ A ρ)

∂ ρ

ρ

∂ A ϕ

∂ ϕ

∂ Az

∂ z

r

2

∂( r

2 Ar )

∂ r

r sin θ

∂ A θ sin θ

∂ θ

r sin θ

∂ A ϕ

∂ ϕ

Curl ∇ × A

i j k

∂ x

∂ y

∂ z

Ax Ay Az

ρ

ρ ϕ̂

ρ

z

∂ ρ

∂ ϕ

∂ z

A ρ ρ A ϕ Az

r

2 sin θ

r

r sin θ

θ

r

ϕ̂

∂ r

∂ θ

∂ ϕ

Ar rA θ rA ϕ sin θ

Laplacian

2 φ

2 φ

∂ x

2

2 φ

∂ y

2

2 φ

∂ z

2

ρ

∂ ρ

ρ

∂ φ

∂ ρ

ρ

2

2 φ

∂ ϕ

2

2 φ

∂ z

2

r

2

∂ r

r

2 ∂ φ

∂ r

r

2 sin θ

∂ θ

sin θ

∂ φ

∂ θ

r

2 sin

2 θ

2 φ

∂ ϕ

2

Transformation of integrals

L = the distance along some curve ‘C’ in space and is measured from some fixed point.

S = a surface area

τ = a volume contained by a specified surface

̂ t = the unit tangent to C at the point P

̂ n = the unit outward pointing normal

A = some vector function

d L = the vector element of curve (= ̂ t d L )

d S = the vector element of surface (= ̂ n d S )

Then

Z

C

A · ̂ t d L =

Z

C

A · d L

and when A = ∇ φ Z

C

(∇ φ) · d L =

Z

C

d φ

Gauss’s Theorem (Divergence Theorem)

When S defines a closed region having a volume τ Z

τ

(∇ · A ) d τ =

Z

S

( A · ̂ n ) d S =

Z

S

A · d S

also Z

τ

(∇ φ) d τ =

Z

S

φ d S

Z

τ

(∇ × A ) d τ =

Z

S

(̂ n × A ) d S

Stokes’s Theorem

When C is closed and bounds the open surface S ,

Z

S

(∇ × A ) · d S =

Z

C

A · d L

also Z

S

(̂ n × ∇ φ) d S =

Z

C

φ d L

Green’s Theorem

Z

S

ψ∇ φ · d S =

Z

τ

∇ · ( ψ∇ φ) d τ

Z

τ

[

ψ∇

2 φ + (∇ ψ) · (∇ φ)

]

d τ

Green’s Second Theorem

Z

τ

( ψ∇

2 φ − φ∇

2 ψ) d τ =

Z

S

[ ψ(∇ φ) − φ(∇ ψ)] · d S

5. Complex Variables

Complex numbers

The complex number z = x + i y = r (cos θ + i sin θ) = r e

i( θ+ 2 n π) , where i

2 = −1 and n is an arbitrary integer. The

real quantity r is the modulus of z and the angle θ is the argument of z. The complex conjugate of z is z

∗ = x − i y =

r (cos θ − i sin θ) = r e

−i θ ; zz

∗ = | z |

2 = x

2

• y

2

De Moivre’s theorem

(cos θ + i sin θ)

n = e

i n θ = cos n θ + i sin n θ

Power series for complex variables.

e

z = 1 + z +

z

2

z

n

n!

• · · · convergent for all finite z

sin z = z −

z

3

z

5

− · · · convergent for all finite z

cos z = 1 −

z

2

z

4

− · · · convergent for all finite z

ln( 1 + z ) = z −

z

2

z

3

− · · · principal value of ln( 1 + z )

This last series converges both on and within the circle | z | = 1 except at the point z = −1.

tan

− 1 z = z −

z

3

z

5

This last series converges both on and within the circle | z | = 1 except at the points z = ±i.

( 1 + z )

n = 1 + nz +

n ( n − 1 )

z

2

n ( n − 1 )( n − 2 )

z

3

• · · ·

This last series converges both on and within the circle | z | = 1 except at the point z = −1.

7. Hyperbolic Functions

cosh x =

( e

x

• e

− x ) = 1 +

x

2

x

4

• · · · valid for all x

sinh x =

( e

x − e

− x ) = x +

x

3

x

5

• · · · valid for all x

cosh i x = cos x cos i x = cosh x

sinh i x = i sin x sin i x = i sinh x

tanh x =

sinh x

cosh x

sech x =

cosh x

coth x =

cosh x

sinh x

cosech x =

sinh x

cosh

2 x − sinh

2 x = 1

For large positive x :

cosh x ≈ sinh x →

e

x

tanh x → 1

For large negative x :

cosh x ≈ − sinh x →

e − x

tanh x → − 1

Relations of the functions

sinh x = − sinh(− x ) sech x = sech(− x )

cosh x = cosh(− x ) cosech x = − cosech(− x )

tanh x = − tanh(− x ) coth x = − coth(− x )

sinh x =

2 tanh ( x / 2 )

1 − tanh

2 ( x / 2 )

tanh x √

1 − tanh

2 x

cosh x =

1 + tanh

2 ( x / 2 )

1 − tanh

2 ( x / 2 )

1 − tanh

2 x

tanh x =

1 − sech

2 x sech x =

1 − tanh

2 x

coth x =

cosech

2 x + 1 cosech x =

coth

2 x − 1

sinh( x / 2 ) =

cosh x − 1

cosh( x / 2 ) =

cosh x + 1

tanh( x / 2 ) =

cosh x − 1

sinh x

sinh x

cosh x + 1

sinh( 2 x ) = 2 sinh x cosh x tanh( 2 x ) =

2 tanh x

1 + tanh

2 x

cosh( 2 x ) = cosh

2 x + sinh

2 x = 2 cosh

2 x − 1 = 1 + 2 sinh

2 x

sinh( 3 x ) = 3 sinh x + 4 sinh

3 x cosh 3 x = 4 cosh

3 x − 3 cosh x

tanh( 3 x ) =

3 tanh x + tanh

3 x

1 + 3 tanh

2 x

sinh( x ± y ) = sinh x cosh y ± cosh x sinh y

cosh( x ± y ) = cosh x cosh y ± sinh x sinh y

tanh( x ± y ) =

tanh x ± tanh y

1 ± tanh x tanh y

sinh x + sinh y = 2 sinh

( x + y ) cosh

( x − y ) cosh x + cosh y = 2 cosh

( x + y ) cosh

( x − y )

sinh x − sinh y = 2 cosh

( x + y ) sinh

( x − y ) cosh x − cosh y = 2 sinh

( x + y ) sinh

( x − y )

sinh x ± cosh x =

1 ± tanh ( x / 2 )

1 ∓ tanh( x / 2 )

= e

± x

tanh x ± tanh y =

sinh( x ± y )

cosh x cosh y

coth x ± coth y = ±

sinh( x ± y )

sinh x sinh y

Inverse functions

sinh

− 1 x

a

= ln

x +

x 2

• a 2

a

for −∞ < x < ∞

cosh

− 1 x

a

= ln

x +

x^2 − a^2

a

for x ≥ a

tanh

− 1 x

a

ln

a + x

a − x

for x

2 < a

2

coth

− 1 x

a

ln

x + a

x − a

for x

2 > a

2

sech

− 1 x

a

= ln

a

x

a

2

x

2

 (^) for 0 < x ≤ a

cosech

− 1 x

a

= ln

a

x

a

2

x

2

for x 6 = 0

8. Limits

n

c x

n → 0 as n → ∞ if | x | < 1 (any fixed c )

x

n / n! → 0 as n → ∞ (any fixed x )

( 1 + x / n )

n → e

x as n → ∞, x ln x → 0 as x → 0

If f ( a ) = g ( a ) = 0 then lim x → a

f ( x )

g ( x )

f

′ ( a )

g

′ ( a )

(l’H ˆopital’s rule)

Z ∞

0

( 1 + x ) x

p d x^ =^ π^ cosec^ p π^ for^ p^ <^1

Z ∞

0

cos( x

2 ) dx =

Z ∞

0

sin( x

2 ) d x =

π

Z (^) ∞

−∞

exp(− x

2 / 2 σ

2 ) d x = σ

2 π

Z ∞

−∞

x

n exp(− x

2 / 2 σ

2 ) d x =

1 × 3 × 5 × · · · ( n − 1 ) σ

n + 1

2 π

for n ≥ 2 and even

for n ≥ 1 and odd Z

sin x d x = − cos x + c

Z

sinh x d x = cosh x + c Z

cos x d x = sin x + c

Z

cosh x d x = sinh x + c Z

tan x d x = − ln(cos x ) + c

Z

tanh x d x = ln(cosh x ) + c Z

cosec x d x = ln(cosec x − cot x ) + c

Z

cosech x d x = ln [tanh( x / 2 )] + c Z

sec x d x = ln(sec x + tan x ) + c

Z

sech x d x = 2 tan

− 1 ( e

x ) + c

Z

cot x d x = ln(sin x ) + c

Z

coth x d x = ln(sinh x ) + c

Z

sin mx sin nx d x =

sin( m − n ) x

2 ( m − n )

sin( m + n ) x

2 ( m + n )

• c if m

2 6 = n

2

Z

cos mx cos nx d x =

sin( m − n ) x

2 ( m − n )

sin( m + n ) x

2 ( m + n )

• c if m

2 6 = n

2

Standard substitutions

If the integrand is a function of: substitute:

( a

2 − x

2 ) or

a 2 − x 2 x = a sin θ or x = a cos θ

( x

2

• a

2 ) or

x^2 + a^2 x = a tan θ or x = a sinh θ

( x

2 − a

2 ) or

x 2 − a 2 x = a sec θ or x = a cosh θ

If the integrand is a rational function of sin x or cos x or both, substitute t = tan( x / 2 ) and use the results:

sin x =

2 t

1 + t

2

cos x =

1 − t

2

1 + t

2

d x =

2 d t

1 + t

2

If the integrand is of the form: substitute:

Z d x

( ax + b )

px + q

px + q = u

2

Z d x

( ax + b )

px 2

• qx + r

ax + b =

u

Integration by parts

Z (^) b

a

u d v = uv

b

a

Z (^) b

a

v d u

Differentiation of an integral

If f ( x , α) is a function of x containing a parameter α and the limits of integration a and b are functions of α then

d

d α

Z b ( α)

a ( α)

f ( x , α) d x = f ( b , α)

d b

d α

− f ( a , α)

d a

d α

Z b ( α)

a ( α)

∂ α

f ( x , α) d x.

Special case,

d

d x

Z (^) x

a

f ( y ) d y = f ( x ).

Dirac δ -‘function’

δ( t − τ) =

2 π

Z (^) ∞

−∞

exp[i ω( t − τ)] d ω.

If f ( t ) is an arbitrary function of t then

Z ∞

−∞

δ( t − τ) f ( t ) d t = f ( τ).

δ( t ) = 0 if t 6 = 0, also

Z ∞

−∞

δ( t ) d t = 1

Reduction formulae

Factorials

n! = n ( n − 1 )( n − 2 )... 1, 0! = 1.

Stirling’s formula for large n : ln( n !) ≈ n ln n − n.

For any p > −1,

Z ∞

0

x

p e

− x d x = p

Z ∞

0

x

p − 1 e

− x d x = p !. (− 1 / 2 )! =

π, ( 1 / 2 )! =

√ π/ 2 , etc.

For any p , q > −1,

Z 1

0

x

p ( 1 − x )

q d x =

p! q!

( p + q + 1 )!

Trigonometrical

If m , n are integers,

Z π/ 2

0

sin

m θ cos

n θ d θ =

m − 1

m + n

Z π/ 2

0

sin

m − 2 θ cos

n θ d θ =

n − 1

m + n

Z π/ 2

0

sin

m θ cos

n − 2 θ d θ

and can therefore be reduced eventually to one of the following integrals

Z π/ 2

0

sin θ cos θ d θ =

Z π/ 2

0

sin θ d θ = 1,

Z π/ 2

0

cos θ d θ = 1,

Z π/ 2

0

d θ =

π

Other

If In =

Z (^) ∞

0

x

n exp(− α x

2 ) d x then In =

( n − 1 )

2 α

In − 2 , I 0 =

π

α

, I 1 =

2 α

Laplace’s equation

2 u = 0

If expressed in two-dimensional polar coordinates (see section 4), a solution is

u ( ρ, ϕ) =

[

A ρ

n

• B ρ

− n

][

C exp(i n ϕ) + D exp(−i n ϕ)

]

where A , B , C , D are constants and n is a real integer.

If expressed in three-dimensional polar coordinates (see section 4) a solution is

u ( r , θ, ϕ) =

[

Ar

l

• Br

−( l + 1 )

]

P

m l

[

C sin m ϕ + D cos m ϕ

]

where l and m are integers with l ≥ | m | ≥ 0; A , B , C , D are constants;

P

m l (cos θ) = sin

| m | θ

[

d

d(cos θ)

]

| m |

Pl (cos θ)

is the associated Legendre polynomial.

P

0 l (^1 ) =^ 1.

If expressed in cylindrical polar coordinates (see section 4), a solution is

u ( ρ, ϕ, z ) = Jm ( n ρ)

[

A cos m ϕ + B sin m ϕ

][

C exp( nz ) + D exp(− nz )

]

where m and n are integers; A , B , C , D are constants.

Spherical harmonics

The normalized solutions Y

m l (^ θ,^ ϕ)^ of the equation [ 1

sin θ

∂ θ

sin θ

∂ θ

sin

2 θ

2

∂ ϕ

2

]

Y

m l

• l ( l + 1 ) Y

m l

are called spherical harmonics, and have values given by

Y

m l ( θ, ϕ) =

2 l + 1

4 π

( l − | m |)!

( l + | m |)!

P

m l (cos θ) e

i m ϕ ×

m for m ≥ 0

1 for m < 0

i.e., Y

0 0

4 π

, Y

0 1 =

4 π

cos^ θ,^ Y

± 1 1 =^ ∓

8 π

sin^ θ^ e

±i ϕ , etc.

Orthogonality

Z

4 π

Y

∗ m l Y

m ′ l ′^ d^ Ω^ =^ δ ll ′ δ mm ′

12. Calculus of Variations

The condition for I =

Z b

a

F ( y , y

′ , x ) d x to have a stationary value is

∂ F

∂ y

d

d x

∂ F

∂ y

′

, where y

′

d y

d x

. This is the

Euler–Lagrange equation.

13. Functions of Several Variables

If φ = f ( x , y , z ,.. .) then

∂ φ

∂ x

implies differentiation with respect to x keeping y , z ,... constant.

dφ =

∂ φ

∂ x

d x +

∂ φ

∂ y

d y +

∂ φ

∂ z

d z + · · · and δ φ ≈

∂ φ

∂ x

δ x +

∂ φ

∂ y

δ y +

∂ φ

∂ z

δ z + · · ·

where x , y , z ,... are independent variables.

∂ φ

∂ x

is also written as

∂ φ

∂ x

y ,...

or

∂ φ

∂ x

y ,...

when the variables kept

constant need to be stated explicitly.

If φ is a well-behaved function then

2 φ

∂ x ∂ y

2 φ

∂ y ∂ x

etc.

If φ = f ( x , y ),

∂ φ

∂ x

y

∂ x

∂ φ

y

∂ φ

∂ x

y

∂ x

∂ y

φ

∂ y

∂ φ

x

Taylor series for two variables

If φ( x , y ) is well-behaved in the vicinity of x = a , y = b then it has a Taylor series

φ( x , y ) = φ( a + u , b + v ) = φ( a , b ) + u

∂ φ

∂ x

• v

∂ φ

∂ y

u

2 φ

∂ x

2

• 2 uv

2 φ

∂ x ∂ y

• v

2 φ

∂ y

2

where x = a + u , y = b + v and the differential coefficients are evaluated at x = a , y = b

Stationary points

A function φ = f ( x , y ) has a stationary point when

∂ φ

∂ x

∂ φ

∂ y

= 0. Unless

2 φ

∂ x

2

2 φ

∂ y

2

2 φ

∂ x ∂ y

= 0, the following

conditions determine whether it is a minimum, a maximum or a saddle point.

Minimum:

2 φ

∂ x

2

> 0, or

2 φ

∂ y

2

Maximum:

2 φ

∂ x

2

< 0, or

2 φ

∂ y

2

and

2 φ

∂ x

2

2 φ

∂ y

2

2 φ

∂ x ∂ y

Saddle point:

2 φ

∂ x

2

2 φ

∂ y

2

2 φ

∂ x ∂ y

If

2 φ

∂ x

2

2 φ

∂ y

2

2 φ

∂ x ∂ y

= 0 the character of the turning point is determined by the next higher derivative.

Changing variables: the chain rule

If φ = f ( x , y ,.. .) and the variables x , y ,... are functions of independent variables u , v ,... then

∂φ

∂ u

∂ φ

∂ x

∂ x

∂ u

∂ φ

∂ y

∂ y

∂ u

∂φ

∂ v

∂ φ

∂ x

∂ x

∂ v

∂ φ

∂ y

∂ y

∂ v

etc.