ENGINEERING

MATHEMATICS FORMULAS &

SHORT NOTES HANDBOOK

Vector Algebra
2

If i, j, k are orthonormal vectors and A = A x i + A y j + A z k then | A| = A2x + A2y + A2z . [Orthonormal vectors
orthogonal unit vectors.]

Scalar product
A B = | A| | B| cos

where is the angle between the vectors

Bx
= A x Bx + A y B y + A z Bz = [ A x A y A z ] B y
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 + b

with a real number.

Page 1 of 24

Equation of a plane
A point r ( x, y, z) is on a plane if either

(a) r b
d = |d|, where d is the normal from the origin to the plane, or
y
z
x
(b) + + = 1 where X, Y, Z are the intercepts on the axes.
X
Y
Z

Vector product
A B = n | A| | B| sin , where is the angle between the vectors and n is a unit vector normal to the plane containing
A and B in the direction for which A, B, n form a right-handed set of axes.

A B in determinant form


 i
j
k 

 Ax A y Az 


 Bx B y Bz 

A B in matrix form

0
Az A y
Bx
Az
0
Ax By
A y Ax
0
Bz

Vector multiplication is not commutative: A B = B A.

Scalar triple product


 Ax

A B C = A B C =  Bx
 Cx

Ay
By
Cy

Vector triple product


A z 
Bz  = A C B,
Cz 

A ( B C ) = ( A C ) B ( A B)C,

etc.

( A B) C = ( A C ) B ( B C ) A

Non-orthogonal basis
A = A1 e1 + A2 e2 + A3 e3
A1 = 0 A

where 0 =

Similarly for A2 and A3 .

e2 e3
e1 (e2 e3 )

Summation convention
a

= ai ei

ab

= ai bi

implies summation over i = 1 . . . 3

where 123 = 1;

( a b)i = i jk a j bk
i jkklm = il jm im jl

Page 2 of 24

i jk = ik j

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 = 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
l

Aik Bk j

( AB)i j =

k=1

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 AA1 = A1 A = I.

( A1 )i j =

transpose of cofactor of A i j
| A|

where the cofactor of A i 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 jk... A1i A2 j A3k . . .

i, j,k,...

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

22 matrices
If A =

a
c

b
d

then,

| A| = ad bc

AT =

a
b

c
d

A1 =

1
| A|

d
c

b
a

Product rules
( AB . . . N ) T = N T . . . B T A T
( AB . . . N )1 = N 1 . . . B1 A1

(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 q i form a set of orthonormal vectors. For any orthogonal
matrix Q,
Q1 = Q T ,

| Q| = 1,

Q T is also orthogonal.

Page 3 of 24

Solving sets of linear simultaneous equations

If A is square then Ax = b has a unique solution x = A 1 b if A1 exists, i.e., if | A| 6= 0.

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

An over-constrained set of equations Ax = 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 | Ax b|) is the solution of the n equations A T Ax = 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 i and eigenvectors u i of an n n matrix A are the solutions of the equation Au = u. The
eigenvalues are the zeros of the polynomial of degree n, Pn ( ) = | A I |. If A is Hermitian then the eigenvalues
i are real and the eigenvectors u i are mutually orthogonal. | A I | = 0 is called the characteristic equation of the
matrix A.
Tr A = i ,
i

also | A| =

i .
i

If S is a symmetric matrix, 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 =

and

S = UU T.

If x is an approximation to an eigenvector of A then x T Ax/( x T x) (Rayleighs quotient) is an approximation to the

corresponding eigenvalue.

Commutators
[ A, B]
[ A, B]
[ A, B]

AB BA
= [ B, A]

= [ 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 = (b1 , b2 , . . .)
Matrix form
Hermiticity

b A c = ( A b) c

Eigenvalues, real

Au i = (i) ui

Orthogonality
Completeness

Operator form
Z

O =

Bra-ket form

(O)

ui u j = 0
b = ui (ui b)

= i

i j = 0

Z

h|O|i
O |i i = i | i i

Oi = (i)i
Z

hi | j i = 0
= |i i hi |i
i

RayleighRitz
Lowest eigenvalue

b A b
0
b b

0 Z

Page 4 of 24

h|O|i
h|i

(i 6 = j )

Pauli spin matrices

x =

0
1


1
,
0

x y = i z ,

y =

0
i

y z = i x ,


i
,
0

z =

z x = i y ,

1
0

0
1

x x = y y = z z = I

Vector Calculus
Notation
is a scalar function of a set of position coordinates. In Cartesian coordinates
= ( x, y, z); in cylindrical polar coordinates = (, , z); in spherical
polar coordinates = (r, , ); in cases with radial symmetry = (r).
A is a vector function whose components are scalar functions of the position
coordinates: in Cartesian coordinates A = iA x + jA y + kA z , where A x , A y , A z
are independent functions of x, y, z.

+ j +k
In Cartesian coordinates (del) i

y
x
y
z

grad = ,

div A = A,

curl A = A

Identities
grad(1 + 2 ) grad 1 + grad 2

grad(12 ) 1 grad 2 + 2 grad 1

curl( A + A ) curl A1 + curl A2

div( A) div A + (grad ) A,

div( A1 + A2 ) div A1 + div A2

curl( A) curl A + (grad ) A

div( A1 A2 ) A2 curl A1 A1 curl A2

curl( A1 A2 ) A1 div A2 A2 div A1 + ( A2 grad) A1 ( A1 grad) A2

div(curl A) 0,

curl(grad ) 0

curl(curl A) grad(div A) div(grad A) grad(div A) 2 A

grad( A1 A2 ) A1 (curl A2 ) + ( A1 grad) A2 + A2 (curl A1 ) + ( A2 grad) A1

Page 5 of 24

Grad, Div, Curl and the Laplacian

Cartesian Coordinates

Conversion to
Cartesian
Coordinates

Cylindrical Coordinates

x = cos

Vector A

Axi + A y j + Az k

Gradient

i+
j+
k
x
y
z

Divergence

y = sin

b+
b+
b



 i
j k 

 




 x y z 


 Ax A y Az 



1
1 

b
b
b





 z 


 A A A z 

2
2
2
+ 2 + 2
2
x
y
z

b + A
b
Arbr + A

b + A
b + Azb
A
z

1 A
A z
1 ( A )
+
+


z

Laplacian

x = r cos sin y = r sin sin

z = r cos

z=z

A y
A z
A x
+
+
x
y
z

Curl A

Spherical Coordinates

1
1 b

b
br +
+

r
r
r sin

1 A sin
1 (r 2 Ar )
+
2
r
r sin

r
1 A
+
r sin


1
1 b
1


b 
 2
br

 r sin r sin

r








Ar
rA
rA sin 




1
1

2
r
+ 2
sin
r

r2 r
r sin




1 2
2

+ 2 2+ 2

Transformation of integrals
L = the distance along some curve C in space and is measured from some fixed point.
S = a surface area

= a volume contained by a specified surface

bt = the unit tangent to C at the point P
b
n = the unit outward pointing normal
A = some vector function

dL = the vector element of curve (= bt dL)

dS = the vector element of surface (= b

n dS)
Z

Then

A bt dL =

and when A =
Z

() dL =

A dL

Gausss Theorem (Divergence Theorem)

When S defines a closed region having a volume

also

( A) d =

() d =

(A b
n) dS =

dS

A dS
Z

( A) d =

Page 6 of 24

(b
n A) dS

2
1
2
r sin 2
2

Stokess Theorem
When C is closed and bounds the open surface S,
Z

also
Z

( A) dS =

(b
n ) dS =

A dL

dL

Greens Theorem
Z

dS =

() d

Z 

Greens Second Theorem

Z


2 + () () d

(2 2 ) d =

[() ()] dS

Complex Variables
Complex numbers
The complex number z = x + iy = r(cos + i sin ) = r ei( +2n), where i2 = 1 and n is an arbitrary integer. The
real quantity r is the modulus of z and the angle is the argument of z. The complex conjugate of z is z = x iy =
2
r(cos i sin ) = r ei ; zz = | z| = x2 + y2

De Moivres theorem
(cos + i sin )n = ein = cos n + i sin n

Power series for complex variables.

ez

z3
3!
z2
cos z
=1
2!
z2
ln(1 + z) = z
2

sin z

z2
zn
+ +
+
2!
n!
z5
+

5!
z4
+

4!
z3
+

=1+z+
=z

convergent for all finite z

convergent for all finite z
convergent for all finite z
principal value of ln (1 + z)

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

z3
z5
+

3
5
This last series converges both on and within the circle | z| = 1 except at the points z = i.
tan1 z

=z

n(n 1) 2 n(n 1)(n 2) 3

z +
z +
2!
3!
This last series converges both on and within the circle | z| = 1 except at the point z = 1.

(1 + z)n = 1 + nz +

Page 7 of 24

Trigonometric Formulae
cos2 A + sin 2 A = 1

sec2 A tan2 A = 1

cos 2A = cos 2 A sin 2 A

sin 2A = 2 sin A cos A

cosec2 A cot2 A = 1
2 tan A
tan 2A =
.
1 tan2 A

sin ( A B) = sin A cos B cos A sin B

cos A cos B =

cos( A + B) + cos( A B)
2

cos( A B) = cos A cos B sin A sin B

sin A sin B =

cos( A B) cos( A + B)
2

sin A cos B =

sin( A + B) + sin ( A B)
2

tan( A B) =

tan A tan B
1 tan A tan B

sin A + sin B = 2 sin

A+B
AB
cos
2
2

cos2 A =

1 + cos 2A
2

sin A sin B = 2 cos

A+B
AB
sin
2
2

sin 2 A =

1 cos 2A
2

cos A + cos B = 2 cos

A+B
AB
cos
2
2

cos3 A =

3 cos A + cos 3A
4

sin 3 A =

3 sin A sin 3A
4

cos A cos B = 2 sin

AB
A+B
sin
2
2

Relations between sides and angles of any plane triangle

In a plane triangle with angles A, B, and C and sides opposite a, b, and c respectively,
a
b
c
=
=
= diameter of circumscribed circle.
sin A
sin B
sin C
a2 = b2 + c2 2bc cos A
a = b cos C + c cos B

b2 + c2 a2
2bc
ab
C
AB
=
cot
tan
2
a+b
2
q
1
1
1
area = ab sin C = bc sin A = ca sin B = s(s a)(s b)(s c),
2
2
2
cos A =

where s =

Relations between sides and angles of any spherical triangle

In a spherical triangle with angles A, B, and C and sides opposite a, b, and c respectively,
sin a
sin b
sin c
=
=
sin A
sin B
sin C
cos a = cos b cos c + sin b sin c cos A
cos A = cos B cos C + sin B sin C cos a

Page 8 of 24

1
( a + b + c)
2

Hyperbolic Functions
x2
x4
1 x
( e + e x ) = 1 +
+
+
2
2!
4!
1
x3
x5
sinh x = ( ex e x ) = x +
+
+
2
3!
5!

valid for all x

cosh x =

valid for all x

cosh ix = cos x

cos ix = cosh x

sinh ix = i sin x
sinh x
tanh x =
cosh x
cosh x
coth x =
sinh x

sin ix = i sinh x
1
sech x =
cosh x
1
cosech x =
sinh x

cosh 2 x sinh 2 x = 1

For large positive x:

cosh x sinh x

ex
2

tanh x 1

For large negative x:

cosh x sinh x

e x
2

tanh x 1

Relations of the functions

sinh x

= sinh ( x)

sech x

cosh x

= cosh ( x)

cosech x = cosech( x)

tanh x

= tanh( x)

coth x

sinh x =

tanh x

2 tanh ( x/2)
2

1 tanh ( x/2)

=q

tanh x
1 tanh2 x

1 sech x

cosh x =

sech x

= sech( x)

= coth ( x)
1 + tanh2 ( x/2)
2

1 tanh ( x/2)

cosech 2 x + 1
r
cosh x 1
sinh ( x/2) =
2
cosh x 1
sinh x
tanh( x/2) =
=
sinh x
cosh x + 1

cosech x =

sinh (2x) = 2 sinh x cosh x

tanh(2x) =

coth x

1 tanh2 x

= q

coth 2 x 1
r
cosh x + 1
cosh ( x/2) =
2

2 tanh x
1 + tanh 2 x

cosh (2x) = cosh 2 x + sinh 2 x = 2 cosh2 x 1 = 1 + 2 sinh 2 x

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

3 tanh x + tanh3 x

cosh 3x = 4 cosh 3 x 3 cosh x

1 + 3 tanh2 x

Page 9 of 24

1
1 tanh2 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

1
sinh x + sinh y = 2 sinh ( x + y) cosh
2
1
sinh x sinh y = 2 cosh ( x + y) sinh
2
sinh x cosh x =
tanh x tanh y =

1
( x y)
2
1
( x y)
2

1 tanh ( x/2)
= e x
1 tanh( x/2)
sinh ( x y)
cosh x cosh y

coth x coth y =

Inverse functions

sinh ( x y)
sinh x sinh y

!
x2 + a2
sinh
a
!
p
x + x2 a2
1 x
cosh
= ln
a
a


1
a+x
1 x
tanh
= ln
a
2
ax


x
1
x
+a
1
coth
= ln
a
2
xa

s
2
x
a
a
sech1 = ln +
1
a
x
x2

s
2
a
a
x
+ 1
cosech1 = ln +
a
x
x2
1

x
= ln
a

1
1
cosh x + cosh y = 2 cosh ( x + y) cosh ( x y)
2
2
1
1
cosh x cosh y = 2 sinh ( x + y) sinh ( x y)
2
2

x+

for < x <

for x a
for x2 < a2
for x2 > a2
for 0 < x a
for x 6= 0

Limits
nc xn 0 as n if | x| < 1 (any fixed c)
xn /n! 0 as n (any fixed x)

(1 + x/n)n ex as n , x ln x 0 as x 0
If f ( a) = g( a) = 0

12

then

lim

x a

f 0 ( a)
f ( x)
= 0
g( x)
g ( a)

(lHopitals

rule)

Page 10 of 24

Differentiation
(uv)0 = u0 v + uv0 ,

 u 0
v

u0 v uv0
v2

(uv)(n) = u(n) v + nu(n1) v(1) + + n Cr u(nr) v(r) + + uv(n)

 
n!
n
n
=
where Cr
r!(n r)!
r
d
(sin x) = cos x
dx
d
(cos x) = sin x
dx
d
(tan x) = sec2 x
dx
d
(sec x) = sec x tan x
dx
d
(cot x) = cosec2 x
dx
d
(cosec x) = cosec x cot x
dx

d
(sinh x)
dx
d
(cosh x)
dx
d
(tanh x)
dx
d
(sech x)
dx
d
(coth x)
dx
d
(cosech x)
dx

Leibniz Theorem

= cosh x
= sinh x
= sech2 x
= sech x tanh x
= cosech2 x
= cosech x coth x

Integration
Standard forms
Z

xn dx =

1
dx
x
Z
eax dx
Z

Z
Z
Z
Z
Z
Z
Z
Z
Z
Z

xn+1
+c
n+1

= ln x + c

for n 6= 1
Z

ln x dx = x(ln x 1) + c


1
x
ax
ax
x e dx = e
2 +c
a
a

1 ax
e +c
a


x2
1
x ln x dx =
ln x
+c
2
2
x
1
1
dx = tan1
+c
2
2
a
a
a +x


 
a+x
1
1
1
1 x
tanh
ln
dx
=
+
c
=
+c
a
a
2a
ax
a2 x2


 
1
1
1
xa
1 x
dx = coth
+c=
+c
ln
a
a
2a
x+a
x2 a2
x
1
1
dx =
+c
2(n 1) ( x2 a2 )n1
( x2 a2 )n
x
1
dx = ln( x2 a2 ) + c
2
x2 a2
x
1
p
dx = sin1
+c
a
a2 x2


p
1
p
dx = ln x + x2 a2 + c
x2 a2
p
x
p
dx = x2 a2 + c
x2 a2
 x i
p
1h p 2
a2 x2 dx =
x a x2 + a2 sin 1
+c
2
a

Page 11 of 24

for x2 < a2
for x2 > a2
for n 6= 1

Z
Z

1
dx = cosec p
(1 + x) x p

cos( x ) dx =

for p < 1

1
sin ( x ) dx =
2
2

exp( x2 /2 2 ) dx = 2

Z
1 3 5 (n 1) n+1 2
n
2
2
x exp( x /2 ) dx =

Z
Z
Z
Z
Z

sin x dx

cos x dx
tan x dx

cot x dx

for n 1 and odd

sinh x dx

= cosh x + c

= sin x + c

cosh x dx

= sinh x + c

= ln(cos x) + c

tanh x dx

= ln(cosh x) + c

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

= ln(sec x + tan x) + c

sech x dx

= 2 tan1 ( ex ) + c

= ln(sin x) + c

coth x dx

= ln(sinh x) + c

= cos x + c

cosec x dx = ln(cosec x cot x) + c

sec x dx

for n 2 and even

sin (m + n) x
sin (m n) x

+c
2(m n)
2(m + n)
Z
sin (m + n) x
sin (m n) x
+
+c
cos mx cos nx dx =
2(m n)
2(m + n)
Z

sin mx sin nx dx =

if m2 6= n2
if m2 6= n2

Standard substitutions
If the integrand is a function of:
p
( a2 x2 ) or a2 x2
p
( x2 + a2 ) or x2 + a2
p
( x2 a2 ) or x2 a2

substitute:
x = a sin or x = a cos
x = a tan or x = a sinh
x = a sec or x = a cosh

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 =

2t
1 + t2

cos x =

1 t2
1 + t2

If the integrand is of the form:

Z
Z

dx
p
( ax + b) px + q

dx
q
( ax + b) px2 + qx + r

dx =

2 dt
.
1 + t2

substitute:
px + q = u2

ax + b =

1
.
u

Page 12 of 24

Integration by parts
Z

b
a

b Z b

u dv = uv
v du
a
a

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

d
d

a ( )

f ( x, ) dx = f (b, )

db
da
f ( a, )
+
d
d

Z b( )

f ( x, ) dx.

a ( )

Special case,
d
dx

x
a

f ( y) dy = f ( x).

Dirac -function
1
(t ) =
2

exp[i(t )] d.

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

(t) = 0 if t 6= 0, also

(t ) f (t) dt = f ( ).

(t) dt = 1

Reduction formulae
Factorials
n! = n(n 1)(n 2) . . . 1,

0! = 1.

Stirlings formula for large n:

For any p > 1,

For any p, q > 1,

x p e x dx = p
1

ln(n!) n ln n n.
Z

x p (1 x)q dx =

x p1 e x dx = p!.

( 1/2)! =

( 1/2)! =

/ ,
2

p!q!
.
( p + q + 1)!

Trigonometrical
If m, n are integers,
m 1 / 2
n 1 / 2
sin m2 cosn d =
sin m cosn2 d
m+n 0
m+n 0
0
and can therefore be reduced eventually to one of the following integrals
Z / 2

sin m cos n d =

Z / 2

sin cos d =

1
,
2

Z / 2
0

sin d = 1,

Z / 2
0

cos d = 1,

Z / 2
0

Other
If In =

xn exp( x2 ) dx

then

In =

(n 1)
In 2 ,
2

I0 =

Page 13 of 24

1
2

I1 =

1
.
2

d =

.
2

etc.

Differential Equations
Diffusion (conduction) equation

= 2
t

Wave equation
2 =

1 2
c2 t2

Legendres equation
(1 x2 )

dy
d2 y
2x
+ l (l + 1) y = 0,
dx2
dx

1
solutions of which are Legendre polynomials Pl ( x), where Pl ( x) = l
2 l!
1
2
P0 ( x) = 1, P1 ( x) = x, P2 ( x) = (3x 1) etc.
2
Recursion relation
Pl ( x) =

1
[(2l 1) xPl 1 ( x) (l 1) Pl 2( x)]
l

Orthogonality
Z

Pl ( x) Pl 0 ( x) dx =

2
ll 0
2l + 1

Bessels equation
x2

d2 y
dy
+x
+ ( x2 m2 ) y = 0,
dx2
dx

solutions of which are Bessel functions Jm ( x) of order m.

Series form of Bessel functions of the first kind

(1)k ( x/2)m+2k
k!(m + k)!
k=0

Jm ( x ) =

(integer m).

The same general form holds for non-integer m > 0.

Page 14 of 24

d
dx

l

l
x2 1 , Rodrigues formula so

Laplaces equation
2 u = 0

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




u(, ) = An + Bn C exp(in) + D exp(in)

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, , ) = Arl + Br(l +1) Plm C sin m + D cos m
where l and m are integers with l |m| 0; A, B, C, D are constants;

| m |
d
Plm (cos ) = sin|m|
Pl (cos )
d(cos )

is the associated Legendre polynomial.

Pl0 (1) = 1.

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




u(, , z) = Jm (n) A cos m + B sin m C exp(nz) + D exp(nz)

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

Spherical harmonics
The normalized solutions Ylm ( , ) of the equation




1

2
1
sin
+
Ylm + l (l + 1)Ylm = 0
sin

sin2 2
are called spherical harmonics, and have values given by
s

m
2l + 1 (l |m|)! m
for m 0
m
im
Yl ( , ) =
(1)
Pl (cos ) e
4 (l + |m|)!
1
for m < 0
r
r
r
1
3
3
0
1
0
i.e., Y0 =
, Y1 =
cos , Y1 =
sin ei , etc.
4
4
8
Orthogonality
Z

Ylm Ylm0 d = ll 0 mm0

0

Calculus of Variations
The condition for I =

b
a

F ( y, y0, x) dx to have a stationary value is

EulerLagrange equation.

Page 15 of 24

F
d
=
y
dx


dy
F
0
. This is the
0 , where y =
dx
y

Functions of Several Variables

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

x

d =
dx +
dy +
dz + and
x +
y +
z +
x
y
z
x
y
z

 


is also written as
when the variables kept
where x, y, z, . . . are independent variables.
or
x
x
x 

If = f ( x, y, z, . . .) then

y,...

constant need to be stated explicitly.

If is a well-behaved function then

If = f ( x, y),
 

1
=   ,
x
x y
y

y,...

2
2
=
etc.
x y
y x

 
y

x
y

 

= 1.

Taylor series for two variables

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

2
2 

1
2
2
+
( x, y) = ( a + u, b + v) = ( a, b) + u
u
+ 2uv
+v
+
+v
x
y
2!
x y
x2
y2

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

y=b

Stationary points
2
2

2
=
=
= 0. Unless 2 =
= 0, the following
2
x
y
x y
x
y
conditions determine whether it is a minimum, a maximum or a saddle point.

2
2

> 0, or
> 0,
 2 2
Minimum:

2 2

x2
y2
and
>
2
2
2
2

x
y

x y

Maximum:
< 0, or
< 0,

2
2
x
y

2
2
2 2
Saddle point:
<
x y
x2 y2

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

If

2
2
2
=
=
= 0 the character of the turning point is determined by the next higher derivative.
x y
x2
y2

Changing variables: the chain rule

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

x
y
=
+
+
u
x u
y u
x
y

=
+
+
v
x v
y v
etc.

Page 16 of 24

Changing variables in surface and volume integrals Jacobians

If an area A in the x, y plane maps into an area A 0 in the u, v plane then


 x x 


Z
Z
 u v 


f ( x, y) dx dy =
f (u, v) J du dv where J = 

A
A0
 y y 


u v
( x, y)
The Jacobian J is also written as
. The corresponding formula for volume integrals is
(u, v)


 x x x 


 u v w 


Z
Z
 y y y 

f ( x, y, z) dx dy dz =
f (u, v, w) J du dv dw
where now
J = 

V
V0
 u v w 


 z z z 


u v w

Fourier Series and Transforms

Fourier series
If y( x) is a function defined in the range x then
y( x) c0 +

cm cos mx +

m=1

M0

sm sin mx

m=1

where the coefficients

are
Z
1
y( x) dx
c0 =
2
Z
1
cm =
y( x) cos mx dx

Z
1
sm =
y( x) sin mx dx

(m = 1, . . . , M)
(m = 1, . . . , M 0 )

with convergence to y( x) as M, M 0 for all points where y( x) is continuous.

Fourier series for other ranges

Variable t, range 0 t T, (i.e., a periodic function of time with period T, frequency = 2/ T).
y(t) c0 + cm cos mt + sm sin mt

where

T
T
T
y(t) dt, cm =
y(t) cos mt dt, sm =
y(t) sin mt dt.
2 0
0
0
Variable x, range 0 x L,
2mx
2mx
y( x) c0 + cm cos
+ sm sin
L
L
where
Z
Z
Z
2 L
1 L
2 L
2mx
2mx
dx, sm =
dx.
c0 =
y( x) dx, cm =
y( x) cos
y( x) sin
L 0
L 0
L
L 0
L
c0 =

Page 17 of 24

Fourier series for odd and even functions

If y( x) is an odd (anti-symmetric) function [i.e., y( x) = y( x)] defined in the range x , then only
Z
2
sines are required in the Fourier series and s m =
y( x) sin mx dx. If, in addition, y( x) is symmetric about
0
Z
4 / 2
y( x) sin mx dx (for m odd). If
x = /2, then the coefficients s m are given by sm = 0 (for m even), s m =
0
y( x) is an even (symmetric) function [i.e., y( x) = y( x)] defined in the range x , then only constant
Z
Z
2
1
y( x) dx, cm =
y( x) cos mx dx. If, in
and cosine terms are required in the Fourier series and c 0 =
0
0

addition, y( x) is anti-symmetric about x = , then c0 = 0 and the coefficients c m are given by cm = 0 (for m even),
2
Z
4 / 2
cm =
y( x) cos mx dx (for m odd).
0
[These results also apply to Fourier series with more general ranges provided appropriate changes are made to the
limits of integration.]

Complex form of Fourier series

If y( x) is a function defined in the range x then
M

y( x)

Cm eimx ,

Cm =

1
2

y( x) eimx dx

with m taking all integer values in the range M. This approximation converges to y( x) as M under the same
conditions as the real form.
For other ranges the formulae are:
Variable t, range 0 t T, frequency = 2/ T,

y(t) =

Cm e

im t

Cm =
2

Variable x0 , range 0 x0 L,

y( x ) =

Cm e

i2mx0 / L

1
Cm =
L

y(t) eimt dt.

L
0

y( x0 ) ei2mx / L dx0 .
0

Discrete Fourier series

If y( x) is a function defined in the range x which is sampled in the 2N equally spaced points x n =
nx/ N [n = ( N 1) . . . N ], then
y( xn ) = c0 + c1 cos xn + c2 cos 2xn + + c N 1 cos( N 1) xn + c N cos Nxn

+ s1 sin xn + s2 sin 2xn + + s N 1 sin ( N 1) xn + s N sin Nxn

where the coefficients are

1
y( xn )
c0 =
2N
1
cm =
y( xn ) cos mxn
N
1
cN =
y( xn ) cos Nxn
2N
1
sm =
y( xn ) sin mxn
N
1
y( xn ) sin Nxn
sN =
2N
each summation being over the 2N sampling points x n .

Page 18 of 24

(m = 1, . . . , N 1)

(m = 1, . . . , N 1)

Fourier transforms
If y( x) is a function defined in the range x then the Fourier transform b
y() is defined by the equations
Z
Z
1
b
by() =
y() eit d,
y(t) eit dt.
y(t) =
2

If is replaced by 2 f , where f is the frequency, this relationship becomes

y(t) =

by( f ) ei2 f t d f ,

by( f ) =

y(t) ei2 f t dt.

If y(t) is symmetric about t = 0 then

Z
Z
1
by() cos t d,
by() = 2
y(t) =
y(t) cos t dt.
0
0
If y(t) is anti-symmetric about t = 0 then
Z
Z
1
by() sin t d,
by() = 2
y(t) sin t dt.
y(t) =
0
0

Specific cases

y(t) = a,
= 0,

|t|
|t| >

y(t) = a(1 |t|/ ),

= 0,

y(t) = exp(t2 /t20 )

y(t) = f (t) ei0 t

by() = 2a

(Top Hat),

|t|
|t| >

(Saw-tooth),

m =

where

(modulated function),

by() = bf ( 0 )

(t m ) (sampling function)

by() =

Page 19 of 24

sinc( x) =



2a
2
(
1

cos

)
=
a

sinc
2
2

by() = t0 exp 2 t20 /4

(Gaussian),

y(t) =

by() =

sin
2a sinc ( )

n =

( 2n/ )

sin ( x)
x

Convolution theorem
If z(t) =

x( ) y(t ) d =

x(t ) y( ) d

x(t) y(t) then

Conversely, xcy = b
x by.

bz () = b
x() by().

Parsevals theorem
Z

y (t) y(t) dt =

1
2

by () by() d

(if b
y is normalised as on page 21)

Fourier transforms in two dimensions

b (k) =
V

V (r ) eikr d2 r

2rV (r) J0 (kr) dr

if azimuthally symmetric
Examples

Fourier transforms in three dimensions

b (k) =
V

V (r )

V (r ) eikr d3 r

4
=
V (r) r sin kr dr
k 0
Z
1
b (k) eikr d3 k
V (r ) =
V
(2)3
Z

if spherically symmetric

1
4r
e r
4r
V (r )

2 V (r )

Page 20 of 24

b (k)
V

1
k2
1
2
k + 2
b (k)
ikV
b (k)
k2 V

Laplace Transforms
If y(t) is a function defined for t 0, the Laplace transform y(s) is defined by the equation
y(s) = L{ y(t)} =

est y(t) dt

Function y(t) (t > 0)

Transform y(s)

(t)

Delta function

(t)

1
s

Unit step function

n!
sn+1
r
1
2 s3
r

tn
1

t /2
1

t /2

1
(s + a)

e at
sin t

(s2

s
(s2 + 2 )

(s2 2 )
s
(s2 2 )

cos t
sinh t
cosh t
e at y(t)

y( s + a )

e s y ( s )

y(t ) (t )
ty(t)

dy
dt
dn y
dtn
Z
Z
Z

t
0
t
0

t
0

y( ) d

x( ) y(t ) d

x(t ) y( ) d

+ 2

dy
ds

s y( s ) y ( 0 )
n

s y( s ) s

n1

y(0) s

n2

dy
dt

dn1 y

dtn1
0

y( s )
s

x ( s ) y( s )

[Note that if y(t) = 0 for t < 0 then the Fourier transform of y(t) is by() = y(i).]

Page 21 of 24

Convolution theorem

Numerical Analysis
Finding the zeros of equations
If the equation is y = f ( x) and x n is an approximation to the root then either
f ( xn )
.
xn+1 = xn 0
f ( xn )
xn xn1
or, xn+1 = xn
f ( xn )
f ( xn ) f ( xn1 )

(Newton)
(Linear interpolation)

are, in general, better approximations.

Numerical integration of differential equations

If

dy
= f ( x, y) then
dx
yn+1 = yn + h f ( xn , yn ) where h = xn+1 xn

(Euler method)

yn+1 = yn + h f ( xn , yn )
h[ f ( xn , yn ) + f ( xn+1 , yn+1 )]
yn+1 = yn +
2

Putting
then

(improved Euler method)

Central difference notation

If y( x) is tabulated at equal intervals of x, where h is the interval, then y n+1/2 = yn+1 yn and
2 yn = yn+1/2 yn1/2

Approximating to derivatives


dy
dx

d2 y
dx2

y 1 + yn 1/2
yn+1 yn
yn yn1

n+ /2
h
h
2h

where h = xn+1 xn

2 y n
yn+1 2yn + yn1
=
h2
h2

Interpolation: Everetts formula

y( x) = y( x0 + h) y0 + y1 +

1
1
2
( 1)2 y0 + ( 2 1)2 y1 +
3!
3!

where is the fraction of the interval h (= x n+1 xn ) between the sampling points and = 1 . The first two
terms represent linear interpolation.

Numerical evaluation of definite integrals

Trapezoidal rule
The interval of integration is divided into n equal sub-intervals, each of width h; then


Z b
1
1
f ( x) dx h c f ( a) + f ( x1 ) + + f ( x j ) + + f (b)
2
2
a
where h = (b a)/n and x j = a + jh.
Simpsons rule
The interval of integration is divided into an even number (say 2n) of equal sub-intervals, each of width h =
(b a)/2n; then
Z b

h
f ( a) + 4 f ( x1 ) + 2 f ( x2 ) + 4 f ( x3 ) + + 2 f ( x2n2 ) + 4 f ( x2n1 ) + f (b)
f ( x) dx
3
a

Page 22 of 24

Gausss integration formulae

These have the general form
For n = 2 :
For n = 3 :

xi = 05773;

y( x) dx ci y( xi )
1

c i = 1, 1 (exact for any cubic).

xi = 07746, 00, 07746;

c i = 0555, 0888, 0555 (exact for any quintic).

Treatment of Random Errors

Sample mean

x=

1
( x1 + x2 + xn )
n

Residual:

d=xx
1
Standard deviation of sample:
s = (d21 + d22 + d2n )1/2
n
1
Standard deviation of distribution:
(d21 + d22 + d2n )1/2
n1

1
m = = q
(d21 + d22 + d2n )1/2
Standard deviation of mean:
n
n ( n 1)

Result of n measurements is quoted as x m .

=q

n ( n 1)

x2i

xi

2

1 / 2

Range method
A quick but crude method of estimating is to find the range r of a set of n readings, i.e., the difference between
the largest and smallest values, then
r
.
n
This is usually adequate for n less than about 12.

Combination of errors
If Z = Z ( A, B, . . .) (with A, B, etc. independent) then

2 
2
Z
Z
A +
B +
( Z )2 =
A
B

So if
(i)

Z = A B C,

(ii)

Z = AB or A/ B,

(iii)

Z = Am ,

(iv)

Z = ln A,

(v)

Z = exp A,

( Z )2 = ( A )2 + ( B )2 + (C )2
 2   2   2
Z
B
A
=
+
Z
A
B
Z

=m A
Z
A
A
Z =
A
Z
= A
Z

Page 23 of 24

Statistics
Mean and Variance
A random variable X has a distribution over some subset x of the real numbers. When the distribution of X is
discrete, the probability that X = x i is Pi . When the distribution is continuous, the probability that X lies in an
interval x is f ( x)x, where f ( x) is the probability density function.
Mean = E( X ) = Pi xi or

x f ( x) dx.

Variance 2 = V ( X ) = E[( X )2 ] =

Pi (xi )2 or

( x )2 f ( x) dx.

Probability distributions
Error function:
Binomial:
Poisson:
Normal:

x
2
2
e y dy
erf( x) =
0
 
n x n x
f ( x) =
p q
where q = (1 p),
x

= np, 2 = npq, p < 1.

x
e , and 2 =
x!


1
( x )2
f ( x) = exp
2 2
2
f ( x) =

Weighted sums of random variables

If W = aX + bY then E(W ) = aE( X ) + bE(Y ). If X and Y are independent then V (W ) = a 2 V ( X ) + b2 V (Y ).

Statistics of a data sample x 1 , . . . , xn

1
n

Sample mean x =

xi

1
Sample variance s =
n
2

( x i x )

1
x2
n i

x2 = E( x2 ) [E( x)]2

Regression (least squares fitting)

To fit a straight line by least squares to n pairs of points ( x i , yi ), model the observations by y i = + ( xi x) + i ,
where the i are independent samples of a random variable with zero mean and variance 2 .
Sample statistics: s 2x =

1
n

( x i x ) 2 ,

s2xy

s2y =

1
n

( y i y) 2 ,

s2xy =

1
n

(xi x)( yi y).

n
(residual variance),
n2
s4
1
b ( xi x)}2 = s2 xy .
b
where residual variance = { yi
y
n
s2x

b=
b = y,
Estimators:

s2x

b ( x x); b 2 =
b+
; E(Y at x) =

b2
b 2
b are
b and
Estimates for the variances of
and 2 .
n
ns x
b=r=
Correlation coefficient:

s2xy

sx s y

Page 24 of 24