GENERAL FORMULA SHEET FOR MATHEMATICS
SECTION I
TRIGONOMETRIC IDENTITIES
1.
cos2   sin 2   1
2.
sin( A  B)  sin A cos B  cos A sin B
3.
sin 2 A  2 sin A cos A
4.
1  tan 2   sec 2 
1  cot 2   cosec 2
cos( A  B)  cos A cos B  sin A sin B
cos 2 A  2 cos2 A  1  1  2 sin 2 A
1
2
1
cos A cos B  [cos( A  B)  cos( A  B)]
2
1
sin A sin B   [cos( A  B)  cos( A  B)]
2
sin A cos B  [sin ( A  B)  sin ( A  B)]
THE EXPONENTIAL AND NATURAL LOGARITHMIC FUNCTIONS
If y  e x  exp( x ) then x  log e y  ln y

y  eln y
and
ln e x  x
i.e. exp and ln are inverse functions. NB
ln y is defined only for y  0 .
-2-
ln
NB
ln a n  n ln a
ln ab  ln a  ln b
Laws of Logarithms:
a x  e x ln a
Fa I  ln a  ln b
Hb K
ln
F1 I   ln a
Ha K
for a  0 .
HYPERBOLIC FUNCTIONS
cosh x 
tanh 
1
2
de  e i,
sinh x
cosh x
x
x
sech x 
sinh x 
1
cosh x
1
2
de  e i
x
x
cosech x 
1
sinh x
Identities
cosh 2 x  sinh 2 x  1
sinh ( A  B)  sinh A cosh B  cosh A sinh B
1  tanh 2 x  sech 2 x
cosh( A  B)  cosh A cosh B  sinh Asinh B
COMPLEX NUMBERS
Argand Diagram
Cartesian Form
z  x  iy ,
z  x  iy
x  Re z ,
y  Im z
(complex conjugate)
Polar Form
x  r cos 

y  r sin 
z  r(cos   i sin )  rei
z  r(cos   i sin )  re i
z  r  x 2  y2
(modulus)
y
x
[NB check that  is in the correct quadrant with       (principal value)]
  arg z where tan  
-3-
Powers:
z n  r nein  [r(cos   i sin )]n  r n (cos n  i sin n)
Roots:
n
z z
1
n
1
 r nei(2k )/ n ,
(de Moivre's theorem)
k  0, 1, 2,...
Relationship between trigonometric and hyperbolic functions
cos  
and
1 i
e  e  i  cosh(i ) ,
2
d
i
cos(i)  cosh 
sin  
1 i
1
e  e  i  sinh(i )
2i
i
d
i
sin(i)  i sinh 
DIFFERENTIATION AND INTEGRATION
Elementary Derivatives and Integrals
y  f (x)
dy
 f ( x )
dx
xn
nx n1
x n1
( n  1)
n 1
sin x
cos x
cos x
 sin x
tan x
sec2 x
ex
ex
1
x
ln x
NB
xn
sinh x
cosh x
cosh x
sinh x
tanh x
sech 2 x
The indefinite integrals in the left-hand column each require an arbitrary constant
e.g.
z
1
dx  ln x  C .
x
-4-
Further Standard Derivatives and Integrals
y  f (x)
dy
 f ( x )
dx
sec x
sec x tan x
cosecx
cosecx cot x
cot x
cosec 2 x
Fx I
Ha K
x
arcsinhFI
Ha K
x
arccoshFI
Ha K
1
x
arc tanFI
Ha K
a
1
x
arc tanhFI
Ha K
a
1
arcsin
2
a  x2
1
a2  x 2
1
x 2  a2
1
2
a  x2
1
a  x2
2
Rules for Differentiation
Product:
d
dv
du
(uv )  u  v
dx
dx
dx
Chain Rule:
if y  y(u) and u  u( x ) then
dy dy du

dx du dx
Implicit Differentiation:
Parametric Differentiation:
FI
HK
d u

dx v
Quotient:
use the chain rule
du
dv
u
dx
dx
v2
(change of variable)
d
df dy
[ f ( y )] 
dx
dy dx
x  x(t ), y  y(t ) where 't' is a parameter
dy dy / dt

dx dx / dt
v
-5-
Newton-Raphson Method
Numerical procedure for the solution of f ( x )  0 : the next approximation xn1 is
xn1  xn 
bg
bg
f xn
f  xn
where xn is the current approximation to the root.
Methods of Integration
z
z
z
z
2x2 1
f ( x )dx 
Substitution (change of variable):
f [ x (u)]
dx
du ,
du
where x  x (u)
f ( x )
dx  ln f ( x )  C
f ( x)
Logarithmic integral:
Partial fractions:
e.g.
d
(1  2 x ) x  3 x  4
2
i
dx 
zLM
N
NB divide first if necessary.
z
Integration by parts:
u
z
dv
du
dx  uv  v dx
dx
dx
Numerical Integration
ah
Trapezium rule:
z
a
h
f ( x )dx  [ f (a)  f (a  h)]
2
Repeat for n strips of equal width h 
z
b
f ( x )dx 
a
ba
with ordinates fk  f xk
n
bg
b
g
h
f0  2 f1  f2 ... fn1  fn
2
Simpsons rule: basic estimate for two strips
a2 h
z
a
h
f ( x )dx  [ f (a)  4 f (a  h)  f (a  2h)]
3
Repeat for an even number n of strips of equal width h 
ba
n
O
P
Q
A
Bx  C
 2
dx
1  2 x x  3x  4
-6-
z
b
f ( x )dx 
a
b
gb
g
h
f0  4 f1  f3 ...  2 f2  f4 ...  fn
3
PARTIAL DIFFERENTIATION
Partial Derivatives
If u( x, y) is a function of x and y
u
 ux  derivative of u with respect to x
x
u
 uy  derivative of u with respect to y
y
(y kept constant)
(x kept constant)
Change of Variables (Chain rule)
If x  x(s, t ) , y  y(s, t ) then f ( x, y) becomes F(s, t ) :
F f x f y


s x s y s
F f x f y


.
t x t y t
SERIES
Arithmetic Progression (A.P.): sum of 'n' terms
1
2
1
2
Sn  a  ( a  d )  ( a  2 d ) .... [ a  ( n  1)d ]  n[2 a  ( n  1)d ]  n( a  l )
where l  a  (n  1)d is the last term.
Geometric Progression (G.P.):
Sn  a  ar  ar ... ar
2

S   ar n1 
n 1
a
1 r
n 1
if

d i
a 1 rn
1 r
r 1
(sum of 'n' terms)
(sum to infinity)
Taylor Series:
( x  a )2
( x  a )3
f ( x )  f ( a )  ( x  a ) f ( a ) 
f ( a ) 
f ( a ) ...
2!
3!
expresses f(x) as a power series about the point x  a . (See Section II for the truncated Taylor
series and remainder term.)
-7-
Maclaurin Series:
f ( x )  f ( 0 )  x f ( 0 ) 
x2
x3
f ( 0 ) 
f ( 0 ) ...
2!
3!
This is the special case of the Taylor series when a  0 i.e. a power series expansion for f(x)
about the origin. The following are examples of Maclaurin series.
Binomial Series:
(1  x )n  1  nx 
n( n  1) 2 n( n  1)( n  2 ) 3
x 
x ...
2!
3!
If n is a positive integer, the series terminates and is valid for all x. The coefficient of x r is then
n
n!

the usual Binomial coefficient
.
r
r !( n  r )!
F
IJ
G
HK
If n is not a positive integer, the series is valid for 1  x  1.
Trigonometric series: sin x  x 
cos x  1 
x3 x5 x7


...
3! 5 ! 7 !
2
4
6
x
x
x


...
2! 4! 6!
U
||
V
||
W
x2 x3

...
2 ! 3!
Exponential series:
ex  1 x 
Logarithmic series:
x2 x3 x 4
ln (1  x )  x 


...
2
3
4
(valid for all x).
(valid for all x).
(valid for 1  x  1) .
-8-
VECTORS
Components
b
a  a1i  a2 j  a3k  a1, a2 , a3
g
a  a12  a22  a32
Scalar Product
a  b  a b cos   a1b1  a2 b2  a3b3

, the vectors are perpendicular
2
or orthogonal and a  b  0 .
If  
Vector Product
i j k
a  b  a b sin n  a1 a2 a3
b1 b2 b3
where n is a unit vector normal (i.e. orthogonal) to a and b
such that (a, b, n) is a right-hand set. Hence, a  b is
perpendicular to both a and b.
CONIC SECTIONS
Circle:
( x  p)2  ( y  q)2  r
Ellipse:
x 2 y2

1
a2 b2
centre ( p, q) , radius r
-9-
Hyperbola:
x 2 y2

1
a2 b2
with asymptotes y  
Rectangular hyperbola:
b
x
a
xy  c
with asymptotes x  0 and y  0
y2  4ax
Parabola:
x  at 2
parametric equations
y  2at
COMPLETING THE SQUARE
ax
2
2
L
O
F
bI F
b 2  4ac I
M
 bx  c  a G
x  J  G 2 JP
M
H 2a K H 4a KP
N
Q
- 10 -
SECTION II
VECTORS (See also Section I for elementary vectors)
Vector Dynamics
Derivatives of unit radial r and transverse  vectors:
d
r  
dt
d 
  r
dt
Moment M of a force F about P is
M  dF
Velocity v of a point P in a rotating body is
v   d
2-D Motion: radial and transverse components of
velocity
v  r  rr  r 
acceleration
1 d 2 
 r  r 2 r 
r  
r dt
d
i
di
3-D Motion: with the usual notation
velocity
v  V r
acceleration
  r    (  r)
a  A  2  V  
Grad, Div and Curl
Vector operator del:
i



 j k
x
y
z
- 11 Gradient of a scalar field ( x, y, z) :
F


I



 j k J
  i  j k
G
Hx y z K x y z
grad    i
Divergence of a vector field V( x, y, z )  V1i  V2 j  V3k :
F


I
V V V
 j  k J b
V i  V j  V kg



G
Hx y z K
x y
z
divV    V  i
1
2
1
3
2
3
Curl (or rotation) of a vector field V( x, y, z)
i
Fi   j   k  IJ bV i  V j  V kg 
curl V    V  G
Hx y z K
x
1
2
3
V1
j
k

y

z
V2
V3
SERIES
Taylor Series for a function of one variable (See Section I for elementary Taylor series)
f ( x )  f ( a )  ( x  a ) f ( a ) 
( x  a )2
( x  a)n1 (n1)
f ( a)... 
f
( a)  Rn
2!
( n  1)!
where the remainder after 'n' terms is
Rn 
( x  a ) n ( n)
f ()
n!
with  lying between a and x i.e.   a  ( x  a), 0    1 .
Fourier Series
For a function f(x) defined in the interval  L  x  L
f ( x) 
a0 
nx
nx
  an cos
 bn sin
2 n1
L
L
F
H
I
K
z
z
where
L
1
nx
an 
f ( x ) cos
dx
L L
L
L
1
nx
bn 
f ( x )sin
dx .
L L
L
- 12 -
Taylor series for a function of two variables
f ( x, y )  f ( a, b )  ( x  a ) f x  ( y  b ) fy
1
( x  a)2 f xx  2( x  a)( y  b) f xy  ( y  b)2 f yy
2!
1

( x  a)3 f xxx  3( x  a)2 ( y  b) f xxy  3( x  a)( y  b)2 f xyy  ( y  b)3 f yyy ...
3!

n
n
s
s
f
f
2 f
where fx  , fy  , f xy 
, ... are all evaluated at the point ( x, y)  (a, b) .
x
y
xy
STATIONARY POINTS OF f (x, y)
The function f ( x, y) has a stationary point at (a, b) if
f
0
x
and
f
0
y
at
( a, b )
Identifying the Stationary Point (a, b)
I.
Maximum/Minimum Point
di
fxx fyy  fxy
II.
2
and

fxx and
fyy  0 MAXIMUM

fxx and
fyy  0
Saddle Point
di
fxx fyy  fxy
III.
2
If
di
fxx fyy  fxy
2
then further investigation is necessary.
MINIMUM
- 13 -
NUMERICAL SOLUTION OF DIFFERENTIAL EQUATIONS
1.
Approximate solution of the first-order equation
bg
y   f ( x, y ) ,
y x0  y0
Euler Method:
'k'-notation
b g
yn1  yn  hf xn , yn
b g
yn1  yn  k
k  hf xn , yn
ynp1  yn  k1
k1  hf xn , yn
Improved Euler Method
b g
h
 fb
x ,y g
 fe
x
2
ynp1  yn  hf xn , yn
yn 1  yn
2.
n
n
p
n 1 , yn 1
j
yn 1  yn 
k2
bk  k g
1
n 1
p
n 1
2
The above methods can be extended to the pair of first-order equations
bg
zb
x g
z
y   f ( x, y, z ) ,
y x0  y0 ,
z   g( x, y, z ) ,
e.g.
1
2
b g
 hf e
x ,y j
0
0
.
b g
 hgb
x ,y ,z g
yn 1  yn  hf xn , yn , zn
Euler Method:
zn1  zn
n
n
n
Improved Euler (using 'k' notation):
ynp1  yn  k1
znp1  zn  l1
3.
b g
l  hgb
x ,y ,z g
k  hf e
x ,y ,z j
l  hge
x ,y ,z j
k1  hf xn , yn , zn
1
bk  k g
bl  l g
yn 1  yn 
1
2
zn 1  zn 
1
2 1
1
2
2
n
n
n
2
n 1
p
n 1
p
n 1
2
n 1
p
n 1
p
n 1
Extension to the second-order equation
y  g( x, y, y) ,
bg
y x0  y0 ,
bg
y x0  y0
Let y  z , y  z and write as the pair of first-order equations [and use (2) with f ( x, y, z)  z ]
y  z ,
z  g( x, y, z) ,
bg
bg
y x0  y0 ,
z x0  y0 .
- 14 -
LAPLACE TRANSFORMS
The Laplace transform of the function f (t ) is
z

k p
F(s)  e  st f (t )dt  L f (t )
0
provided the integral exists.
Standard Laplace Transforms
f (t )
L



1
L
1
t
t n n  1, 2, 3,...
e at
sin at
cos at
sinh at
cosh at
F( s )
1
s
1
s2
n!
s n 1
1
sa
a
2
s  a2
s
2
s  a2
a
2
s  a2
s
2
s  a2
(t )
1
(t  a)
e as
H (t  a)
e as
s
- 15 -
Properties of Laplace Transforms
1.
Linear operator:
L{af (t)  bg(t )}  aL{ f (t )}  bL{g(t )}
2.
First shift theorem:
L e f (t )  F ( s  a )
3.
Derivatives:
L{ f (t )}  sF(s)  f (0)
at
{
}
2
L{ f (t )]  s F(s)  sf (0)  f (0)
R
U
F( s )
f
(
u
)
du

S
V
z
T Ws
t
4.
Integral:
L
0
L{t f (t )}  
dF( s)
ds
5.
Multiplication by t:
6.
Second shift theorem: L{H(t  a) f (t  a)}  e
 as
F(s)