VISVESVARAYA TECHNOLOGICAL UNIVERSITY
BELAGAVI
MATHEMATICS HANDBOOK
I and II Semester BE Program
Effective from the academic year 2022-2023
VISVESVARAYA TECHNOLOGICAL UNIVERSITY, BELAGAVI
MATHEMATICS HANDBOOK
Derivatives of some standard functions:
 
d n
x  n x n 1
dx
d
c  0
dx
 
 
d x
e  ex
dx
d x
a  a x log a
dx
d
 sin x   cos x
dx
d
 cos x    sin x
dx
d
 tan x   sec2 x
dx
d
 cot x    cos ec 2 x
dx
d
 sec x   sec x tan x
dx
d
 cos ecx    cos ecx cot x
dx
d
1
 log x  
dx
x
log e
d
 log a x   a
dx
x


d
1
cos 1 x 
dx
1  x2
d
1
tan 1 x 
dx
1  x2


d
1
cot 1 x 
dx
1  x2
d
 sinh x   cosh x
dx
d
 cosh x   sinh x
dx
d
 tanh x   sec h2 x
dx
d
 coth x    cos ech2 x
dx
d
 sec hx    sec hx tanh x
dx
d
 cos echx    cos echx coth x
dx
d
1
sin 1 x 
dx
1  x2




Rules of Differentiation:
𝑑
𝑑𝑢
(𝑐𝑢) = 𝑐
𝑑𝑥
𝑑𝑥
d
d
d
 uv   u  v   v  u 
dx
dx
dx
1|Page
d u
 
dx  v 
v
d
d
u   u v 
dx
dx
v2
VISVESVARAYA TECHNOLOGICAL UNIVERSITY, BELAGAVI
MATHEMATICS HANDBOOK
Parametric differentiation:
If 𝑥 = 𝑥(𝑡) & 𝑦 = 𝑦(𝑡) 𝑡ℎ𝑒𝑛
𝑑𝑦
𝑑𝑥
=
𝑑𝑦
)
𝑑𝑡
𝑑𝑥
( )
𝑑𝑡
(
Chain Rule:
If 𝑦 = 𝑓(𝑢) & 𝑢 = 𝑔(𝑥) then
𝑑𝑦
𝑑𝑥
=
𝑑𝑦 𝑑𝑢
𝑑𝑢 𝑑𝑥
Integrals of some standard functions:
(Constant of Integration C to be added in all the integrals)
x n 1
1
n
 x dx  n  1
 xdx  log x
 log xdx  x log x  x, x  0
 kdx  k x
ax
log a
x
x
 e dx  e
x
 a dx 
 sin x dx   cos x
 cos x dx  sin x
 tan x dx  log  sec x 
 cot x dx  log  sin x 
 sec x dx  log  sec x  tan x 
 cos ecx dx  log  cos ecx  cot x 
 sec
 cos ec x dx   cot x
2
x dx  tan x
2
 sec x tan x dx  sec x
 cos ecx
 sinh x dx  cosh x
 cosh x dx  sinh x
 tanh x dx  log  cosh x 
 coth x dx  log  sinh x 
 sec h x dx  tanh x
 cos ech x dx   coth x
2
a
f
2
'
1
1
dx  tan 1  x a 
2
x
a
 x
 f  x  dx  log  f  x  
ax
 e cos bx dx 
ax
 e sin bx dx 
2|Page
eax
 a cos bx  b sin bx 
a 2  b2
eax
 a sin bx  b cos bx 
a 2  b2
cot x dx   cos ecx
2


1
a x
f '  x
2
f  x
2
dx  sin 1  x a 
dx  2 f  x 
VISVESVARAYA TECHNOLOGICAL UNIVERSITY, BELAGAVI
MATHEMATICS HANDBOOK
a

b
a
f  x  dx   f  a  x  dx
0
a

a
a
f  x  dx    f  x  dx

a
0
 a
2 f  x  dx
f  x  dx   0
 0

b
f  x  is even function
if
f  x  is odd function
if
𝑎
2𝑎
2 ∫ 𝑓(𝑥)𝑑𝑥, 𝑖𝑓 𝑓(2𝑎 − 𝑥) = 𝑓(𝑥)
∫ 𝑓(𝑥)𝑑𝑥 = {
0
0
0,
𝑖𝑓 𝑓(2𝑎 − 𝑥) = −𝑓(𝑥)
Integration by parts:
 u  x  v  x  dx  u  x    v  x  dx   
d
u  x 
dx
  v  x  dx  dx
Bernoulli’s rule of integration:
If the 1st function is a polynomial and integration of 2nd function is known. Then
∫ 𝑢(𝑥)𝑣(𝑥)𝑑𝑥 = 𝑢 ∫ 𝑣 𝑑𝑥 − 𝑢′ ∬ 𝑣 𝑑𝑥𝑑𝑥 + 𝑢" ∭ 𝑣 𝑑𝑥𝑑𝑥𝑑𝑥 − 𝑢′′′ ∬ ∬ 𝑣 𝑑𝑥𝑑𝑥𝑑𝑥𝑑𝑥
+ ⋯⋯⋯⋯
Where dashes denote the differentiation of u .
Or
∫ 𝑢(𝑥)𝑣(𝑥)𝑑𝑥 = 𝑢 ∙ 𝑣1 − 𝑢′ ∙ 𝑣2 + 𝑢′′ ∙ 𝑣3 − 𝑢′′′ ∙ 𝑣4 + ⋯ ⋯ ⋯ ⋯
Where dashes denote the differentiation of u , vk denotes the integration of v, k times with respect
to x.
Vector calculus formulae:

Position vector
r  x iˆ  y ˆj  z kˆ

Magnitude
r  x2  y 2  z 2
iˆ  iˆ  ˆj  ˆj  kˆ  kˆ  1 and
Cross product of unit vectors iˆ  iˆ  ˆj  ˆj  kˆ  kˆ  0 and
Dot product of unit vectors
Angle between two vectors
Unit vector
3|Page
cos  
A
Aˆ 
A
A. B
A B
iˆ  ˆj  ˆj  kˆ  kˆ  iˆ  0
iˆ  ˆj  kˆ, ˆj  kˆ  iˆ, iˆ  kˆ   ˆj
VISVESVARAYA TECHNOLOGICAL UNIVERSITY, BELAGAVI
MATHEMATICS HANDBOOK

ds
dt

d 2s
a 2
dt
V
Velocity
Acceleration
⃗⃗ = (𝑎2 𝑖 + 𝑏2 𝑗 + 𝑐2 𝑘) & 𝐶⃗ = (𝑎3 𝑖 + 𝑏3 𝑗 + 𝑐3 𝑘)
For any vectors 𝐴⃗ = (𝑎1 𝑖 + 𝑏1 𝑗 + 𝑐1 𝑘), 𝐵

Dot product of two vectors

A  B  a1 a2  b1 b2  c1 c2
ˆj kˆ
iˆ


Cross product of two vectors A  B  a1
a2
a3
b1
b2
b3
  
A  B C


Scalar triple product
a1
   
   A  B   C  b1
 

c1
a2
a3
b2
b3
c2
c3
Trigonometric formulae:

Identities
sin 2   cos 2   1
1  tan 2   sec2 
1  cot 2   cos e c 2 
 Compound angle formulae
sin  A  B   sin A cos B  cos A sin B
sin  A  B   sin A cos B  cos A sin B
cos  A  B   cos A cos B  sin A sin B
cos  A  B   cos A cos B  sin A sin B
tan A  tan B
1  tan A tan B
tan A  tan B
tan  A  B  
1  tan A tan B
tan  A  B  

Transformation formulae
1
sin  A  B   sin  A  B   ,
2
1
cos A cos B  cos  A  B   cos  A  B   ,
2
CD
CD ,
sin C  sin D  2sin 
 cos 

 2 
 2 
CD
CD ,
cos C  cos D  2 cos 
 cos 

 2 
 2 
sin A cos B 
4|Page
1
sin  A  B   sin  A  B  
2
1
sin A sin B  cos  A  B   cos  A  B  
2
CD
CD
sin C  sin D  2sin 
 cos 

 2 
 2 
CD CD
cos C  cos D  2sin 
 sin 

 2   2 
cos A sin B 
VISVESVARAYA TECHNOLOGICAL UNIVERSITY, BELAGAVI
MATHEMATICS HANDBOOK

Multiple angle formulae
sin 2 A  2 sin A cos A
sin A  2sin  A 2  cos  A 2 
cos 2 A  cos 2 A  sin 2 A
cos A  cos 2  A 2   sin 2  A 2 
sin 2 A 
1  cos 2 
2
1
sin 3 A  3sin A  sin 3 A
4
Hyperbolic and Euler’s formulae
e x  e x
sinh x 
2
cos 2 A 
1  cos 2 
2
1
cos3 A  3cos A  cos 3 A
4
cosh x 
e x  e x
2
cos x 
eix  e ix
2
cosh 2   sinh 2   1
ei  cos   i sin 
sin x 
eix  e ix
2i
Logarithmic formulae:
log e  A B   log e  A   log e  B 
log e x n  n log e x
log a a  1
 A
log e    log e  A   log e  B 
B
log e B
log a B 
log e a
log a 1  0
log e 0  
Solid geometry formulae:
Distance between two points (𝑥1 , 𝑦1 , 𝑧1 ) 𝑎𝑛𝑑 (𝑥2 , 𝑦2 , 𝑧2 ) is
 x2  x1    y2  y1    z2  z1 
2
2
l  cos  , m  cos  , n  cos 
a
b
c
, m
, n
.
Direction ratios l 
a 2  b2  c 2
a 2  b2  c 2
a 2  b2  c 2
Direction ratios of a line joining two points (𝑎, 𝑏, 𝑐) = (𝑥2 − 𝑥1 , 𝑦2 − 𝑦1 , 𝑧2 − 𝑧1 )
Direction cosines
5|Page
2
VISVESVARAYA TECHNOLOGICAL UNIVERSITY, BELAGAVI
MATHEMATICS HANDBOOK
nth Derivatives of standard functions
m
mn
D n  ax  b    m  m  1 m  2  ......  m  n  1 ax  b  .a n


n
D n  ax  b    n !a n


Dn  xn   n!
n
 1   1 n !a
D 


n 1
 ax  b   ax  b 
n
n
 1  n  1!a n .
D log  ax  b   
n
 ax  b 
n
D n  a mx   a mx  m log a 
n 1
n
 
D n eax  a n e ax
D n sin  ax  b    a n sin  ax  b  n. / 2 
D n cos  ax  b    a n cos  ax  b  n. / 2 




D n eax sin  bx  c    a 2  b 2
D n eax cos  bx  c    a 2  b 2
n2
 a 

cos  bx  c  n tan  b  
a
eax sin bx  c  n tan 1 b
n2
eax
1
Polar coordinates and polar curves:
Angle between radius vector and tangent
tan   r
d
1 dr
or cot  
dr
r d
Angle of intersection of the curves
 tan 1  tan 2
 1  tan 1 tan 2
1  2  tan 1 
Orthogonal condition
1  2 




2
or
tan 1  tan 2  1,
Pedal equation or p-r equation
P  r sin 
1
1
1
1 𝑑𝑟 2
2
(1
=
+
cot
∅)
=
(1
+
( ) )
𝑝2 𝑟 2
𝑟2
𝑟 2 𝑑𝜃
6|Page
VISVESVARAYA TECHNOLOGICAL UNIVERSITY, BELAGAVI
MATHEMATICS HANDBOOK
Derivative of arc length:
In Polar:
𝑑𝑠
= √1 + ( )
𝑑𝑥
𝑑𝑥
= √1 +
𝑑𝑟
In Parametric:
𝑠𝑖𝑛∅ = 𝑟
𝑑𝑦 2
𝑑𝑠
In Cartesian:
𝑑𝛳
𝑑𝑠
𝑑𝑠
𝑑𝑡
𝑑𝛳 2
1
𝑑𝑠
&
( )
𝑟 2 𝑑𝑟
𝑑𝑦
𝑑𝑠
&
2
𝑑𝜃
𝑑𝑥 2
= √1 + ( )
𝑑𝑦
𝑑𝑟 2
= √𝑟 2 + ( )
𝑑𝛳
2
𝑑𝑥
𝑑𝑦
= √( ) + ( )
𝑑𝑡
𝑑𝑡
𝑑𝑟
& 𝑐𝑜𝑠∅ = 𝑟
𝑑𝑠
Radius of curvature
3
(1+𝑦1 2 )2
In Cartesian form: 𝜌 =
𝑦2
3
In parametric form: 𝜌 =
(𝑥̇ 2 +𝑦̇ 2 )2
𝑥̇ 𝑦̈−𝑦̇ 𝑥̈
3
(𝑟 2 +𝑟1 2 )2
In polar from: 𝜌 = 𝑟 2 −𝑟𝑟
1 +2𝑟1
Pedal Equation: 𝜌 = 𝑟
2
𝑑𝑟
𝑑𝑝
Indeterminate Forms - L’Hospital’s rule:
𝑓′ (𝑥)
𝑓(𝑥)
If 𝑓(𝑎) = 𝑔(𝑎) = 0 , then lim 𝑔(𝑥) = lim 𝑔′ (𝑥)
𝑥→𝑎
𝑥→𝑎
𝑓′ (𝑥)
𝑓(𝑥)
If 𝑓(𝑎) = 𝑔(𝑎) = ∞ , then lim 𝑔(𝑥) = lim 𝑔′ (𝑥)
𝑥→𝑎
1 𝑛
1
lim (1 + 𝑛) = 𝑒
,
𝑠𝑖𝑛𝜃
𝜃→0 𝜃
,
𝑛→∞
lim
𝑥→𝑎
=1
lim (1 + 𝑛)𝑛 = 𝑒
𝑛→0
𝑥 𝑛 −𝑎 𝑛
𝑥→𝑎 𝑥−𝑎
lim
= 𝑛𝑎𝑛−1
Series Expansion:
Taylor’s series expansion about the point x  a.
y  x  y a
 x  a  y'

1!
a
 x  a

2!
2
y a
''
 x  a

3
3!
y '''  a   ..
Maclaurin’s Series at the point x  0
y  x   y  0 
7|Page
x '
x2
x3
x4
y  0   y ''  0   y '''  0   y ''''  0   ....
1!
2!
3!
4!
VISVESVARAYA TECHNOLOGICAL UNIVERSITY, BELAGAVI
MATHEMATICS HANDBOOK
Euler’s theorem on homogeneous function and Corollary:
u
u
x y
n u
(Theorem)
x
y
x
u
u
f (u )
y
n '
x
y
f (u )
𝜕2 𝑢
𝜕2 𝑢
(Corollary 1)
𝜕2 𝑢
𝑥 2 𝜕𝑥 2 + 2𝑥𝑦 𝜕𝑥𝜕𝑦 + 𝑦 2 𝜕𝑦 2 = 𝑛(𝑛 − 1)𝑢
(Corollary 2)
Composite function:
If z  f  x, y  and x    t  , y    t  then
𝑑𝑧
𝑑𝑡
=
𝜕𝑧 𝑑𝑥
𝜕𝑥 𝑑𝑡
+
𝜕𝑧 𝑑𝑦
𝜕𝑦 𝑑𝑡
If z  f  x, y  and x    u , v  , y    u , v 
𝜕𝑧
𝜕𝑢
=
𝜕𝑧 𝜕𝑥
𝜕𝑥 𝜕𝑢
+
𝜕𝑧 𝜕𝑦
𝜕𝑦 𝜕𝑢
𝜕𝑧
&
𝜕𝑣
=
𝜕𝑧 𝜕𝑥
𝜕𝑥 𝜕𝑣
+
𝜕𝑧 𝜕𝑦
𝜕𝑦 𝜕𝑣
If u  f  r , s, t  and r    x, y, z  , s    x, y, z  , t    x, y, z 
u u r u s u t



x r x s x t x
u u r u s u t



y r y s y t y
u u r u s u t



z r z s z t z
Jacobians:
u
  u , v  x
J

  x, y  v
x
u
y
v
y
and
Multiple Integrals:
Area A=
 dx dy - Cartesian form
A
Area A=
 rdrd -Polar form
A
8|Page
u
x
  u , v, w 
v

  x, y , z 
x
w
x
u
y
v
y
w
y
u
z
v
z
w
z
VISVESVARAYA TECHNOLOGICAL UNIVERSITY, BELAGAVI
MATHEMATICS HANDBOOK
Volume V=
 dx dydz - Cartesian form
V
Volume V=
 zdx dy - by double integral
A

Gamma function:   n   e x

x

dx  2 e t t 2 n 1dt
n 1
2
0
0

1
Beta function:   m, n    x n 1 1  x 
m 1
2
dx  2  sin 2 n 1  cos 2 m 1 d
0
0
Beta and Gamma relation:   m, n  
  m   n
 2 
and  1
  m  n

Vector Calculus:


dr
v (t ) 
dt
Velocity



d v d2 r
a (t ) 

.
dt
dt 2
Acceleration
The unit tangent vector


dr
Tˆ 
dt
dr
dt

Angle between the tangents
cos  

T 1 .T 2


T1 T 2

Component of velocity
C.V  v . nˆ
Component of accelerations
C. A  a . nˆ
Tangential component of acceleration
T .C. A  a . v
Normal component of acceleration


N .C. A  a   tangential component    v

, Where n̂ is the unit vector


Gradient of 
grad  =  
 ˆ  ˆ  ˆ
i
j
k
x
z
z
Unit vector normal to the surface

n̂ 

Directional Derivative: D. D   . nˆ
9|Page


v

v


VISVESVARAYA TECHNOLOGICAL UNIVERSITY, BELAGAVI
MATHEMATICS HANDBOOK
Angle between the surfaces
1 .2
cos  
1 2

Divergence of vector field F  f1i  f 2 j  f 3k

. F  divF 
f1 f 2 f3


x z z

Curl of vector field F


  F  curl F 
iˆ
ˆj
kˆ

x
f1

y
f2

z
f3
Solenoidal vector field


div F    F  0
Irrotational vector field


Curl F    F  0 .
List of vector identities
curl ( grad )      0.




div (curl F )       F   0.







div ( F )    div F   grad  F







curl ( F )    curl F   grad  F


Reduction formulae
 /2
 cos
 /2
n
xdx 
0
 /2

0

0
  n  1 n  3 n  5  .........1 
when n is even

 n  n  2  n  4  ..............2 2
n
sin xdx  
  n  1 n  3 n  5  .........3 1 when n is odd
 n  n  2  n  4  ................1

  m  1 m  3 .........  n  1 n  3 .........

  m  n  m  n  2  m  n  4  .............2
m
n
sin x cos xdx  
  m  1 m  3 .........  n  1 n  3 .........
  m  n  m  n  2  m  n  4  ..............

10 | P a g e

2
when m and n is even
other cases
VISVESVARAYA TECHNOLOGICAL UNIVERSITY, BELAGAVI
MATHEMATICS HANDBOOK
Differential Equations:
Differential Equations
dy
Linear in y
 Py  Q
dx
dx
Linear in x
 Px  Q
dy
dy
Bernoulli’s
 P y  Q yn
dx
Exact differential equation
M N
Mdx  Ndy  0 with

.
y
x
Not exact differential equation
Case 1 : If Mdx  Ndy  0 and
Mdx  Ndy  0
Case 2 : If the differential equation is of
the form yg1  xy  dx  xg 2  xy  dy  0
Case 3: If
1  M N 


  f ( x) or C
N  y
x 
Solution/ substitution
y ( I .F )   Q( I .F )dx  C
x  I .F    Q  I .F .dy  C
divide by y n and Put y1 n  z

  terms of N not containing x  dy  C
M dx 
y constant
I .F . 
1
, Mx  Ny  0
Mx  Ny
I .F . 
1
with
Mx  Ny
I .F .  e 
f  x  dx
or e 
Cdx
I .F .  e 
f  y  dy
or e 
Cdy
1  N M 


  f ( y ) or C
M  x y 
Case 5 : If the differential equation is of
the form
'
'
x n y n  mydx  nxdy   x p y q  m' ydx  n' xdy   0
I .F .  x h y k With
Newton’s law of cooling
T  To  C1e  k t
Case 4 : If
Mx  Ny  0
p  h 1 q  k 1
p'  h  1 q'  k  1

and

m
n
m'
n'
Linear Algebra:
Inverse of a square matrix A
 adj A .
A1 
A
Rank of a Matrix A
The number of non-zero rows in the echelon form of A is equal to rank of A
Normal Form of a Matrix.
Ir 0 
0 0 


11 | P a g e
VISVESVARAYA TECHNOLOGICAL UNIVERSITY, BELAGAVI
MATHEMATICS HANDBOOK
Gauss-Elimination Method
The system is reduced to upper triangular system from which the unknowns are found by
back substitutions.
Gauss-Jordan Method
The system is reduced to diagonal system from which the unknowns are found by back
substitutions.
Eigenvalues
Roots of the characteristic equation A   I  0
Eigen Vectors
Non-zero solution x  xi of A   I x  0
Diagonal form
1 0 0 
D  P AP   0 2 0 
 0 0 3 
Computation of power of a square matrix A
1
A n  P D n P 1
Canonical Form
V  1 y 2  2 y 2  3 y 2  .......  n y 2
1
2
3
n
Nature, Rank and Index of Quadratic forms:
 positive-definite if all the eigenvalues of A are positive.
 positive-semi definite if all the eigenvalues of A are non-negative and at least one of the
eigenvalues is zero.
 negative-definite if all the eigenvalues of A are negative.
 negative-semi definite if all the eigenvalues of A are negative and at least one of the
eigenvalues is zero.
 indefinite if the matrix A has both positive and negative eigenvalues.
 Rank the number of non-zero terms
 Index the number of positive terms
 Signature the number of positive terms minus the number of negative terms
12 | P a g e
VISVESVARAYA TECHNOLOGICAL UNIVERSITY, BELAGAVI
MATHEMATICS HANDBOOK
Laplace Transforms:
Laplace Transform of Standard Functions
 
L tn 
1
L 1 
s
n!
, n  1, 2,3.......
s n 1
Where n is a positive integer
 
L e at 
 
1
sa
L e at 
1
sa
L sin at 
a
s  a2
L cos at 
L sinh at 
a
s  a2
L cosh at 
L u  t  
2
2
1
s
s
s  a2
2
s
s  a2
2
e as
L u  t  a  
s
Properties of the Laplace transform:
𝐿{𝑓(𝑡)} = 𝐹(𝑠)
𝑓(𝑡)
Translation
L e at f  t   F  s  a 
(first Shifting Theorem)
13 | P a g e
dn
f  t    1 n F  s 
ds
Multiplication by t
L t
Time scale
1 s
L  f  at   F  
a a
Integration
t



 1
L   f  t  d   L  f  t 


0
 s
Division by t
 f t   
L
   F  s  ds.
 t  s
n
n
VISVESVARAYA TECHNOLOGICAL UNIVERSITY, BELAGAVI
MATHEMATICS HANDBOOK
Transform of a Periodic Function: If f (t) is periodic with a period T , then
T
1
L  f  t  
e st f  t  dt .
 sT 
1  e  0
Second Shifting Theorem:


If F  s   L f  t  and a  0, then L  f  t  a  u  t  a   e  as L  f  t  .
Transforms of the derivatives:
𝐿{𝑓 ′ (𝑡)} = 𝑠𝐿{𝑓(𝑡)} − 𝑓(0)
𝐿{𝑓′′ (𝑡)} = 𝑠 2 𝐿{𝑓(𝑡)} − 𝑠 𝑓(0) − 𝑓′(0)
𝐿{𝑓 𝑛 (𝑡)} = 𝑠 𝑛 𝐿{𝑓(𝑡)} − 𝑠 𝑛−1 𝑓(0) − 𝑠 𝑛−2 𝑓 ′ (0) − 𝑠 𝑛−2 𝑓"(0) − ⋯ ⋯ ⋯ − 𝑓 𝑛−1 (0)
Inverse Laplace Transform:
Transform
F ( s )  L f (t )
 
L tn 
n!
s n 1
 
14 | P a g e
f (t )  L1 F ( s)
t n 1
 1 
 L1  n 
 n  1!
 s 
Where n is a positive integer
L e at 
Inverse Transform
Where n is a positive integer
 1 
eat  L1 

s  a
1
sa
L sinh at 
a
s  a2
 a 
sinh at  L1  2 2 
s  a 
L cosh at 
s
s  a2
 s 
cosh at  L1  2 2 
s  a 
2
2
L sin at 
a
s  a2
 a 
sin at  L1  2 2 
s  a 
L cos at 
s
s  a2
 s 
cos at  L1  2 2 
s  a 
2
2
VISVESVARAYA TECHNOLOGICAL UNIVERSITY, BELAGAVI
MATHEMATICS HANDBOOK
Formulae on Shifting rule:
𝐿[𝑒 𝑎𝑡 𝑐𝑜𝑠𝑏𝑡] =
𝐿[𝑒 𝑎𝑡 𝑠𝑖𝑛𝑏𝑡] =
𝑠−𝑎
(𝑠−𝑎)2 +𝑏2
𝑠+𝑎
+𝑏2
𝑏
𝐿[𝑒 −𝑎𝑡 𝑠𝑖𝑛𝑏𝑡] = (𝑠+𝑎)2
𝐿[𝑒 𝑎𝑡 𝑠𝑖𝑛ℎ𝑏𝑡] =
+𝑏2
𝑏
(𝑠−𝑎)2 −𝑏2
𝑠−𝑎
𝐿[𝑒 𝑎𝑡 𝑐𝑜𝑠ℎ𝑏𝑡] = (𝑠−𝑎)2
−𝑏2
𝑠+𝑎
𝐿[𝑒 −𝑎𝑡 𝑐𝑜𝑠ℎ𝑏𝑡] = (𝑠+𝑎)2
−𝑏2
𝐿
[(𝑠+𝑎)2
+𝑏2
𝑏
𝐿−1 [(𝑠+𝑎)2
𝐿−1 [
= 𝑒 𝑎𝑡 𝑠𝑖𝑛𝑏𝑡
] = 𝑒 −𝑎𝑡 𝑐𝑜𝑠𝑏𝑡
+𝑏2
𝑏
] = 𝑒 −𝑎𝑡 𝑠𝑖𝑛𝑏
] = 𝑒 𝑎𝑡 𝑠𝑖𝑛ℎ𝑏𝑡
(𝑠−𝑎)2 −𝑏2
𝑠−𝑎
𝐿−1 [(𝑠−𝑎)2 2 ]
−𝑏
𝑠+𝑎
−1
𝐿
𝑏
𝐿[𝑒 −𝑎𝑡 𝑠𝑖𝑛ℎ𝑏𝑡] = (𝑠+𝑎)2
] = 𝑒 𝑎𝑡 𝑐𝑜𝑠𝑏𝑡
(𝑠−𝑎)2 +𝑏2
𝑏
𝐿−1 [
]
(𝑠−𝑎)2 +𝑏2
𝑠+𝑎
−1
𝑏
𝐿[𝑒 −𝑎𝑡 𝑐𝑜𝑠𝑏𝑡] = (𝑠+𝑎)2
𝑠−𝑎
𝐿−1 [
(𝑠−𝑎)2 +𝑏2
[(𝑠+𝑎)2
−𝑏2
] = 𝑒 −𝑎𝑡 𝑐𝑜𝑠ℎ𝑏𝑡
𝑏
𝐿−1 [(𝑠+𝑎)2
−𝑏2
= 𝑒 𝑎𝑡 𝑐𝑜𝑠ℎ𝑏𝑡
−𝑏2
] = 𝑒 −𝑎𝑡 𝑠𝑖𝑛ℎ𝑏𝑡
Convolution Theorem:
Let f  t  and g  t  be piecewise continuous on  0,   and F  s   L  f  t  & G  s   L  g  t  .
t
Then, L1 F  s  G  s    f  u g  t  u  du.
0
Numerical Methods:
𝑥𝑘+1 −𝑥𝑘
Regula Falsi formula: 𝑥𝑘+2 = 𝑥𝑘 −
for k = 0,1,2,………
𝑓(𝑥𝑘+1 )−𝑓(𝑥𝑘 )
Newton-Raphson formula:
𝑥𝑛+1 = 𝑥𝑛 −
𝑓(𝑥𝑛 )
𝑓′(𝑥𝑛 )
for n = 1, 2, 3, ……..
Newton’s Forward Interpolation formula:
𝑦𝑝 = 𝑦0 + 𝑝∆𝑦0 +
Where 𝑝 =
𝑝(𝑝−1)
2!
∆2 𝑦0 +
𝑝(𝑝−1)(𝑝−2)
3!
∆3 𝑦0 + ⋯ ⋯ ⋯
𝑥−𝑥0
ℎ
Newton’s Backward Interpolation formula:
𝑦𝑝 = 𝑦0 + 𝑝∇𝑦0 +
Where 𝑝 =
15 | P a g e
𝑥−𝑥𝑛
ℎ
𝑝(𝑝 + 1) 2
𝑝(𝑝 + 1)(𝑝 + 2) 3
∇ 𝑦0 +
∇ 𝑦0 + ⋯ ⋯ ⋯
2!
3!
VISVESVARAYA TECHNOLOGICAL UNIVERSITY, BELAGAVI
MATHEMATICS HANDBOOK
Newton’s General Interpolation formula (Divided difference formula):
𝑦 = 𝑓(𝑥) = 𝑦0 + (𝑥 − 𝑥0 )[𝑥0 , 𝑥1 ] + (𝑥 − 𝑥0 )(𝑥 − 𝑥1 )[𝑥0 , 𝑥1 , 𝑥2 ]
+ (𝑥 − 𝑥0 )(𝑥 − 𝑥1 )(𝑥 − 𝑥2 )[𝑥0 , 𝑥1 , 𝑥2 , 𝑥3 ] + ⋯ ⋯ ⋯
+ (𝑥 − 𝑥0 )(𝑥 − 𝑥1 ) ⋯ ⋯ (𝑥 − 𝑥𝑛 )[𝑥0 , 𝑥1 , ⋯ ⋯ , 𝑥𝑛 ]
Lagrange’s Interpolation formula:
(𝑥 − 𝑥1 )(𝑥 − 𝑥2 )(𝑥 − 𝑥3 ) ⋯ ⋯ ⋯ (𝑥 − 𝑥𝑛 )
𝑓(𝑥) =
𝑦
(𝑥0 − 𝑥1 )(𝑥0 − 𝑥2 )(𝑥0 − 𝑥3 ) ⋯ ⋯ ⋯ (𝑥0 − 𝑥𝑛 ) 0
(𝑥 − 𝑥0 )(𝑥 − 𝑥2 )(𝑥 − 𝑥3 ) ⋯ ⋯ ⋯ (𝑥 − 𝑥𝑛 )
+
𝑦
(𝑥1 − 𝑥0 )(𝑥1 − 𝑥2 )(𝑥1 − 𝑥3 ) ⋯ ⋯ ⋯ (𝑥1 − 𝑥𝑛 ) 1
(𝑥 − 𝑥0 )(𝑥 − 𝑥1 )(𝑥 − 𝑥3 ) ⋯ ⋯ ⋯ (𝑥 − 𝑥𝑛 )
+
𝑦
(𝑥2 − 𝑥0 )(𝑥2 − 𝑥1 )(𝑥2 − 𝑥3 ) ⋯ ⋯ ⋯ (𝑥2 − 𝑥𝑛 ) 2
+⋯⋯⋯⋯
(𝑥 − 𝑥0 )(𝑥 − 𝑥1 )(𝑥 − 𝑥2 ) ⋯ ⋯ ⋯ (𝑥 − 𝑥𝑛 )
+
𝑦
(𝑥𝑛 − 𝑥0 )(𝑥𝑛 − 𝑥1 )(𝑥𝑛 − 𝑥2 ) ⋯ ⋯ ⋯ (𝑥𝑛 − 𝑥𝑛−1 ) 𝑛
Numerical Integration:
Trapezoidal Rule:
𝑥0 +𝑛ℎ
∫
𝑓(𝑥) 𝑑𝑥 =
𝑥0
ℎ
[(𝑦 + 𝑦𝑛 ) + 2(𝑦1 + 𝑦2 + ⋯ ⋯ + 𝑦𝑛−1 )]
2 0
Simpson’s (1/3)rd rule:
𝑥0 +𝑛ℎ
∫
𝑓(𝑥) 𝑑𝑥 =
𝑥0
ℎ
[(𝑦 + 𝑦𝑛 ) + 4(𝑦1 + 𝑦3 + ⋯ + 𝑦𝑛−1 ) + 2(𝑦2 + 𝑦4 + ⋯ + 𝑦𝑛−2 )]
3 0
Simpson’s (3/8)th rule:
𝑥0 +𝑛ℎ
∫
𝑓(𝑥) 𝑑𝑥
𝑥0
3ℎ
[(𝑦0 + 𝑦𝑛 ) + 3(𝑦1 + 𝑦2 + 𝑦4 + 𝑦5 + ⋯ + 𝑦𝑛−1 )
8
+ 2(𝑦3 + 𝑦6 + ⋯ + 𝑦𝑛−3 )]
=
16 | P a g e
VISVESVARAYA TECHNOLOGICAL UNIVERSITY, BELAGAVI
MATHEMATICS HANDBOOK
Numerical methods for ODE’s:
Taylor’s series expansion about the point x  x0 .
y  x   y  x0 
 x  x0  y '

1!
 x0 
 x  x0 

2!
2
y  x0 
''
 x  x0 

3!
3
y '''  x0   ..
Taylor’s series expansion about the point x  0
y  x   y  0 
x '
x2
x3
x4
y  0   y ''  0   y '''  0   y ''''  0   .... Euler’s Method:
1!
2!
3!
4!
y nE1  y n  hf ( x n , y n ),
n  0,1,2,3, 
Modified Euler’s Method:
h
yn 1  yn   f ( xn , yn )  f ( xn 1 , ynE1 )  ,
2
n  0,1, 2,3,
Runge–Kutta Method
k 
h

k 2  hf  xn  , y n  1 
2
2

Step1: Find k1  hf ( x n , y n ) ;
k 
h

k 3  hf  x n  , y n  2  ;
k 4  hf x n  h, y n  k 3 
2
2

1
Step2: Find y n 1  y n  k1  2k 2  2k 3  k 4 
6
Milne’s Method:
Step1: Find f1  f ( x1 , y1 ), f 2  f ( x 2 , y 2 ) and
Step2: Predictor Formula y 4( P )  y 0 
Step3: Compute
4h
2 f1  f 2  2 f 3 
3
f 4  f ( x 4 , y 4( P ) )
Step4: Corrector Formula y4c  y2 
17 | P a g e
f 3  f ( x3 , y 3 ) .

h
f 2  4 f1  f 4P
3

VISVESVARAYA TECHNOLOGICAL UNIVERSITY, BELAGAVI
MATHEMATICS HANDBOOK
Adams – Bashforth Method:
Step1: Find f 0  f  x0 , y0  , f1  f ( x1 , y1 ), f 2  f ( x 2 , y 2 ) and
Step2: Predictor Formula y4P  y3 
Step3: Compute
f 3  f ( x3 , y 3 ) .
h
 55 f3  59 f 2  37 f1  9 f0 
24
f 4  f ( x 4 , y 4( P ) )
Step4: : Corrector Formula y4c  y3 

h
f1  5 f 2  19 f1  9 f 4P
24

Modular Arithmetics:
Set of Natural Numbers (N): {1, 2, 3, …..}
Set of Integers (I) = set of natural, negative naturals and zero: 𝑵 ∪ −𝑵 ∪ {𝟎}
Greatest common divisor: 𝒈 𝒐𝒓 𝒅 = 𝒈𝒄𝒅(𝒂, 𝒃)
Relative Primes: 𝒈 𝒐𝒓 𝒅 = 𝒈𝒄𝒅(𝒂, 𝒃) = 𝟏
Division Algorithm: For integers a and b, with a > 0, there exist integers q and r
such that 𝒃 = 𝒒 ∙ 𝒂 + 𝒓 and 𝟎 ≤ 𝒓 < 𝒂, where q: quotient, r: remainder
6. Euclidean Algorithm
b
a
a
b = q1 a + r1
r 1 = b – a q1
r1
a = q2r1 + r2
r2 = a - q2r1
g = d∙ gcd (a, b) = ax + by with x and y integers
7. Modular and modular class: 𝒎/(𝒂 − 𝒃) then 𝒂 ≡ 𝒃(𝒎𝒐𝒅𝒎), 𝟎 ≤ |𝒃| < 𝒎
𝒃 ≡ 𝒂(𝒎𝒐𝒅𝒎), 𝟎 ≤ |𝒂| < 𝒎
If 𝒂 ≡ 𝒃(𝒎𝒐𝒅𝒎) then 𝒂 − 𝒃 = 𝒌 ∙ 𝒎
8. Properties of modular arithmetic:




𝑎
𝑎
𝑎
𝑎
≡ 𝑎(𝑚𝑜𝑑𝑚)
≡ 𝑏(𝑚𝑜𝑑𝑚) 𝑎𝑛𝑑 𝑏 ≡ 𝑐(𝑚𝑜𝑑𝑚), then 𝑎 ≡ 𝑐(𝑚𝑜𝑑𝑚)
≡ 𝑐(𝑚𝑜𝑑𝑚) 𝑎𝑛𝑑 𝑏 ≡ 𝑐(𝑚𝑜𝑑𝑚), then 𝑎 ≡ 𝑏(𝑚𝑜𝑑𝑚)
≡ 𝑏(𝑚𝑜𝑑𝑚) then for any 𝑐, 𝑎 + 𝑐 ≡ (𝑏 + 𝑐)(𝑚𝑜𝑑𝑚)
18 | P a g e
VISVESVARAYA TECHNOLOGICAL UNIVERSITY, BELAGAVI
MATHEMATICS HANDBOOK



If 𝑎 ≡ 𝑏(𝑚𝑜𝑑𝑚) then for any 𝑐, 𝑎 + 𝑐 ≡ (𝑏 + 𝑐)(𝑚𝑜𝑑𝑚)
If 𝑎 ≡ 𝑏(𝑚𝑜𝑑𝑚) then for any 𝑐, 𝑎 ∙ 𝑐 ≡ (𝑏 ∙ 𝑐)(𝑚𝑜𝑑𝑚)
If 𝑎𝑐 ≡ (𝑏𝑐)(𝑚𝑜𝑑𝑚) 𝑎𝑛𝑑 𝑑 = gcd(𝑚, 𝑐), then 𝑎 ≡ 𝑏(𝑚𝑜𝑑 𝑚/𝑑)

If 𝒂𝒄 ≡ (𝒃𝒄)(𝒎𝒐𝒅𝒎) and m is a prime number, then 𝒂 ≡ 𝒃(𝒎𝒐𝒅 𝒎)


If 𝑎 ≡ 𝑏(𝑚𝑜𝑑𝑚) 𝑎𝑛𝑑 𝑐 ≡ 𝑑(𝑚𝑜𝑑𝑚), then 𝑎 ± 𝑐 ≡ (𝑏 ± 𝑑)(𝑚𝑜𝑑𝑚)
If 𝑎 ≡ 𝑏(𝑚𝑜𝑑𝑚) 𝑎𝑛𝑑 𝑐 ≡ 𝑑(𝑚𝑜𝑑𝑚), then 𝑎 ∙ 𝑐 ≡ (𝑏 ∙ 𝑑)(𝑚𝑜𝑑𝑚)

If 𝒂𝒄 ≡ 𝒃(𝒎𝒐𝒅𝒎) 𝒂𝒏𝒅 𝒄 ≡ 𝒅(𝒎𝒐𝒅𝒎), 𝐭𝐡𝐞𝐧 𝒂𝒃 ≡ 𝒃(𝒎𝒐𝒅𝒎)

If 𝒂 ≡ 𝒃(𝒎𝒐𝒅𝒎), 𝒕𝒉𝒆𝒏 𝒂𝒏 ≡ 𝒃𝒏 (𝒎𝒐𝒅𝒎)

If 𝐠𝐜𝐝(𝒎, 𝒏) = 𝟏, 𝒂 ≡ 𝒃(𝒎𝒐𝒅𝒎) 𝒂𝒏𝒅 𝒂 ≡ 𝒃(𝒎𝒐𝒅𝒏), then 𝒂 ≡ 𝒃(𝒎𝒐𝒅 𝒎𝒏)

If 𝒂𝒃 ≡ 𝟏(𝒎𝒐𝒅 𝒏), 𝒕𝒉𝒆𝒏 𝒂 𝒊𝒔 𝒊𝒏𝒗𝒆𝒓𝒔𝒆 𝒐𝒇 𝒃 𝒂𝒏𝒅 𝒃 𝒊𝒔 𝒕𝒉𝒆 𝒊𝒏𝒗𝒆𝒓𝒔𝒆 𝒐𝒇 𝒂
Linear Diophantine equation:
𝒂𝒙 + 𝒃𝒚 = 𝒄, 𝒘𝒊𝒕𝒉 𝒅 = 𝒈𝒄𝒅(𝒂, 𝒃) 𝒉𝒂𝒔 𝒔𝒐𝒍𝒖𝒕𝒊𝒐𝒏 𝒊𝒇 𝒅/𝒄
(i) has d congruent solutions and if x0 and y0 are primitive solutions,
𝒃
then 𝒙 = 𝒙𝟎 + (𝒅) 𝒕, 𝒚 = 𝒚𝟎 + (𝒂/𝒅)𝒕, for positive integer t
(ii) if d = 1 then it has unique solution
Remainder theorem:
𝒙 ≡ 𝒂𝒊 (𝒎𝒐𝒅 𝒎𝒊 ), 𝒊 = 𝟏, 𝟐, 𝟑, 𝒘𝒊𝒕𝒉 (𝒎𝒊 , 𝒎𝒋 ) = 𝟏, 𝒊 ≠ 𝒋
Procedure to apply remainder theorem
If 𝒎𝒊 , 𝒊 = 𝟏, 𝟐, 𝟑 are relatively prime
Then the solution of 𝒙𝒊 ≡ 𝒃𝒊 (𝒎𝒐𝒅 𝒎𝒊 ), 𝒊 = 𝟏, 𝟐, 3 is 𝒙 = ∑𝟑𝟏 𝒃𝒊 𝑴𝒊 𝒚𝒊
where, Mi = M / mi with 𝑴 = ∏𝟑𝟏 𝒎𝒊
Mi𝒚𝒊 ≡ 𝟏(𝒎𝒐𝒅 𝒎𝒊 ) where 𝒚𝒊 𝒊𝒔 𝒊𝒏𝒗𝒆𝒓𝒔𝒆 𝒐𝒇 Mi under mod mi
11. Fermat’s little theorem:
If p is any prime and p ∤a, then 𝒂𝒑−𝟏 ≡ 1 (mod p) or 𝒂𝒑 ≡ a (mod p)
12. Euler’s Φ function:
If n is any composite number with prime factorization n = 𝒑𝟏 𝒎𝟏 X 𝒑𝟐 𝒎𝟐 𝑿 … ;
then number of primes up to n is
𝟏
𝟏
𝟏
𝟐
𝚽(𝒏) = 𝒏 (𝟏 − 𝒑 ) (𝟏 − 𝒑 ) ⋯ ⋯
13. Wilson’s theorem: Prime p divides (p − 1)! + 1
19 | P a g e
VISVESVARAYA TECHNOLOGICAL UNIVERSITY, BELAGAVI
MATHEMATICS HANDBOOK
14. RSA cryptosystem:
If p and q are large prime numbers and a is any integer then,
plain text M is encrypted to c by, 𝒄 ≡ 𝑴𝒆 (𝒎𝒐𝒅 𝒑𝒒)
The public key will be = (pq, e)
d the decryption key the inverse of e, then 𝒅 ∙ 𝒆 ≡ 𝟏(𝒎𝒐𝒅(𝒑 − 𝟏)(𝒒 − 𝟏))
then 𝑴 ≡ 𝒄𝒅 (𝒎𝒐𝒅𝒑𝒒).
Curvilinear Coordinates:
Curvilinear coordinates are often used to define the location or distribution of physical quantities
which may be scalars, vectors or tensors.
̂ , then
⃗⃗⃗ = 𝒙(𝒖, 𝒗, 𝒘)𝒊̂ + 𝒚(𝒖, 𝒗, 𝒘)𝒋̂ + 𝒛(𝒖, 𝒗, 𝒘)𝒌
If 𝑹
(i)
⃗⃗⃗
𝝏𝑹
⃗⃗⃗
𝝏𝑹
Tangent vectors to the u-curve, the v-curve and the w-curve are 𝝏𝒖 , 𝝏𝒗 𝒂𝒏𝒅
respectively
⃗⃗⃗
𝝏𝑹
⃗⃗⃗
𝝏𝑹
⃗⃗⃗
𝝏𝑹
(ii)
Scale factors 𝒉𝟏 = |𝝏𝒖| , 𝒉𝟐 = |𝝏𝒗 | 𝒂𝒏𝒅 𝒉𝟑 = |𝝏𝒘|
(iii)
Unit Tangent Vectors to the u-curve, the v-curve and the w-curve are
𝑇𝑢 =
⃗⃗⃗
𝜕𝑅
)
𝜕𝑢
(
ℎ1
, 𝑇𝑣 =
⃗⃗⃗
𝜕𝑅
)
𝜕𝑣
(
ℎ2
𝑎𝑛𝑑 𝑇𝑤 =
⃗⃗⃗
𝜕𝑅
)
𝜕𝑤
(
ℎ3
, respectively.
Normal to the surfaces 𝒖 = 𝒖𝟎 , 𝒗 = 𝒗𝟎 𝒂𝒏𝒅 𝒘 = 𝒘𝟎 are
𝑇
𝑇
∇𝑢 = ℎ𝑢 , ∇v = ℎ𝑣 𝑎𝑛𝑑 ∇w =
1
2
𝑇𝑤
ℎ3
respectively
Unit normal vectors to the surfaces 𝒖 = 𝒖𝟎 , 𝒗 = 𝒗𝟎 𝒂𝒏𝒅 𝒘 = 𝒘𝟎 are
∇𝑢
∇v
∇w
𝑁𝑢 = |∇𝑢| , 𝑁𝑣 = |∇v| and 𝑁𝑤 = |∇w| , respectively
Polar Coordinates (𝒓, 𝜽) :
Coordinate transformations: 𝑥 = 𝑟 𝑐𝑜𝑠 𝜃, 𝑦 = 𝑟 𝑠𝑖𝑛 𝜃
Jacobian:
𝜕(𝑥,𝑦)
𝜕(𝑟,𝜃)
=𝑟
(𝐴𝑟𝑐 − 𝑙𝑒𝑛𝑔𝑡ℎ)2 : (𝑑𝑠)2 = (𝑑𝑟)2 + 𝑟 2 (𝑑𝜃)2
20 | P a g e
⃗⃗⃗
𝝏𝑹
𝝏𝒘
VISVESVARAYA TECHNOLOGICAL UNIVERSITY, BELAGAVI
MATHEMATICS HANDBOOK
Cylindrical Coordinates (𝝆, 𝝋, 𝒛):
Coordinate transformations: 𝑥 = 𝜌 𝑐𝑜𝑠 𝜑, 𝑦 = 𝜌 𝑠𝑖𝑛 𝜑, 𝑧 = 𝑧
Jacobian:
𝜕(𝑥,𝑦,𝑧)
𝜕(𝜌,𝜑,𝑧)
=𝜌
(𝐴𝑟𝑐 − 𝑙𝑒𝑛𝑔𝑡ℎ)2 : (𝑑𝑠)2 = (𝑑𝜌)2 + 𝜌2 (𝑑𝜑)2 + (𝑑𝑧)2
Volume element: 𝑑𝑉 = 𝜌𝑑𝜌 𝑑𝜑 𝑑𝑧
Spherical Polar Coordinates (𝒓, 𝜽, 𝝋) :
Coordinate transformations: 𝑥 = 𝑟 sin 𝜃 cos 𝜑 , 𝑦 = 𝑟 sin 𝜃 sin 𝜑 , 𝑧 = 𝑟 cos 𝜃
Jacobian:
𝜕(𝑥,𝑦,𝑧)
𝜕(𝑟,𝜃,𝜑)
= 𝑟 2 sin 𝜃
(𝐴𝑟𝑐 − 𝑙𝑒𝑛𝑔𝑡ℎ)2 : (𝑑𝑠)2 = (𝑑𝑟)2 + 𝑟 2 (𝑑𝜃)2 + (𝑟 sin 𝜃)2 (𝑑𝜑)2
Volume element: 𝑑𝑉 = 𝑟 2 sin 𝜃 𝑑𝑟 𝑑𝜃𝑑𝜑
21 | P a g e