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Engineering Mathematics 2015

NAME OF THE SUBJECT : Mathematics I

SUBJECT CODE : MA6151
NAME OF THE MATERIAL : Formula Material
MATERIAL CODE : HG13AUM101
REGULATION : R2013
UPDATED ON : May-June 2015
(Scan the above Q.R code for the direct download of this material)

Name of the Student: Branch:

Unit I (Matrices)
1. The Characteristic equation of matrix A is
a) 2 S1 S2 0 if A is 2 X 2 matrix

Where S1 Sum of the main diagonal elements.

S2 A

b) 3 S1 2 S2 S3 0 if A is 3 X 3 matrix

Where S1 Sum of the main diagonal elements.

S2 Sum of the minors of the main diagonal elements.
S3 A

2. To find the eigenvectors solve A I X 0 .

3. Property of eigenvalues:
Let A be any matrix then
a) Sum of the eigenvalues = Sum of the main diagonal.
b) Product of the eigenvalues = A

c) If the matrix A is triangular then diagonal elements are eigenvalues.

1
d) If is an eigenvalue of a matrix A, the is the eigenvalue of A1 .

e) If 1 , 2 ,...n are the eigenvalues of a matrix A, then 1m , 2m , ... nm are

eigenvalues of Am .( m being a positive integer)

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Engineering Mathematics 2015

f) The eigenvalues of A & AT are same.

4. Cayley-Hamilton Theorem:
Every square matrix satisfies its own characteristic equation. (ie) A I 0 .

1 1
coeff ( x 12 ) coeff ( x1 x2 ) coeff ( x1 x3 )
2 2
1 1
5. Matrix of the Quadratic form coeff ( x2 x1 ) coeff ( x 22 ) coeff ( x2 x3 )
2 2
1 1
coeff ( x3 x1 ) coeff ( x3 x2 ) coeff ( x 32 )
2 2
6. Index = p = Number of positive eigenvalues
Rank = r = Number of non-zero rows
Signature = s = 2p-r
7. Diagonalisation of a matrix by orthogonal transformation (or) orthogonal
reduction:
Working Rules:
Let A be any square matrix of order n.

Step:1 Find the characteristic equation.

Step:2 Solve the characteristic equation.

Step:3 Find the eigenvectors.

Step:4 Form a normalized model matrix N, such that the eigenvectors are orthogonal.
Step:5 Find N T .
Step:6 Calculate D=NT AN .

Note:
We can apply orthogonal transformation for symmetric matrix only.
If any two eigenvalues are equal then we must use a, b, c method for third eigenvector.

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Engineering Mathematics 2015

Unit II (Sequences and Series)

1. Convergent and Divergent sequence:

If the sequence of real numbers an n1 has a limit L , then the sequence is said

to be a convergent sequence. If it does not have it, then it is said to be divergent.

(i.e) lim an L
n

2. Bounded Sequence:

A Sequence a1 , a2 , a3 ... is bounded if there exist a number M 0 such that

an M , n .

3. Monotone Sequence:

A sequence an is non-decreasing if an an1 for all n and non-increasing if

an an1 for all n . A monotonic sequence is a sequence which is either non-
decreasing or non-increasing.

Example:

A non-decreasing sequence which is bounded above is convergent.

A non-decreasing sequence is always bounded below.
A non-increasing sequence which is bounded below is convergent.
A non-increasing sequence is always bounded above.

4. Comparison Test:

If two series of non-negative terms an and bn such that an bn for all n .
n 1 n 1

Then, if b
n 1
n is convergent then the given series a
n 1
n is convergent.

5. Integral Test:

Consider an integer N and a non-negative function f defined on the

unbounded interval [ N , ) , on which it is monotone decreasing. Then the

infinite series f ( n) converges to a real number if and only if the improper
n N

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Engineering Mathematics 2015

integral f ( x )dx is finite. In other words, if the integral infinite, then the series
N
diverges.

6. D'Alembert's ratio test Ratio Test:

an 1
In a series an of non-negative terms if
n 1

n 1 an
L then the series a
n 1
n is

converges if L 1 , diverges if L 1 and test fails if L 1 .

7. Alternating Series:

A series in which the terms are alternatively positive or negative that is

(1)
n 1
n 1
an a1 a2 a3 ... where an are positive, is called an alternating

series.

8. Leibnitzs Test:

Leibnitzs test is also known as the alternating series test. Given a series

(1)
n 1
n1
an with an 0 , if an is monotonically decreasing as n and

lim an 0 , then the series converges.

n

9. Absolute and Conditional convergent:

An arbitrary series an is called absolutely convergent if a n is convergent.
n 1 n 1

If an is convergent and
n 1
a
n 1
n is divergent we call the series conditionally

convergent.

Unit III (Applications of Differential Calculus)

1. Curvature of a circle = Reciprocal of its radius
3

2. Radius of curvature with Cartesian form

1 y12 2

y2

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3. Radius of curvature if y1 ,
1 x12 2

, where x1
dx
x2 dy

f
3
2
f 2 2
x y
4. Radius of curvature in implicit form
f xx f 2 f xy f x f y f yy f x2
2
y

5. Radius of curvature with parametric form

x 2 y 2 2

xy xy

6. Centre of curvature is x , y .

7. Circle of curvature is x x y y 2 .
2 2

where x x

y1 1 y12 , y y
1 y 2
1

y2 y2

8. Evolute: The locus of centre of curvature of the given curve is called evolute of

the curve. x x

y1 1 y12 , y y
1 y 2
1

y2 y2

9. Envelope: The envelope is a curve which meets each members of a family of

curve.
If the given equation can be rewrite as quadratic equation in parameter, (ie)
A 2 B C 0 where A, B, C are functions of x and y then the envelope is

B2 4 AC 0 .
10. Evolute as the envelope of normals.
Equations Normal equations
y 2 4ax y xt at 3 2at

x 2 4ay x yt at 3 2at

x2 y2 ax by
1 a 2 b2
a 2 b2 cos sin

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Engineering Mathematics 2015

x2 y2 ax by
1 a 2 b2
a 2 b2 sec tan
2 2 2 x cos y sin a cos 2
x3 y3 a3

xy c 2 c
y xt 2 ct 3
t

Unit IV (Differential Calculus of several variables)

1. Eulers Theorem:
If f is a homogeneous function of x and y in degree n , then

f f
(i) x y nf (first order)
x y

2 f 2 f 2 f
2
(ii) x 2 2 xy y n n 1 f (second order)
x 2 xy y 2

du u dx u dy u dz
2. If u f ( x, y, z ) , x g1 (t ), y g2 (t ), z g3 (t ) then
dt x dt y dt z dt

3. If u f ( x, y ), x g1 (r , ), y g2 (r , ) then

u u x u y u u x u y
(i) (ii)
r x r y r x y
4. Maxima and Minima :
Working Rules:
Step:1 Find f x and f y . Put f x 0 and f y 0 . Find the value of x and y.

Step:2 Calculate r f xx , s f xy , t f yy . Now rt s 2

Step:3 i. If 0 , then the function have either maximum or minimum.

1. If r 0 Maximum
2. If r 0 Minimum
ii. If 0, then the function is neither Maximum nor Minimum, it is
called Saddle Point.
iii. If 0, then the test is inconclusive.

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Engineering Mathematics 2015

5. Maxima and Minima of a function using Lagranges Multipliers:

Let f ( x, y, z ) be given function and g( x, y, z ) be the subject to the condition.

Form F ( x, y, z ) f ( x, y, z ) g( x, y, z ) , Putting Fx Fy Fz F 0 and

then find the value of x,y,z. Next we can discuss about the Max. and Min.
6. Jacobian:
u u
u, v ( u, v ) x y
Jacobian of two dimensions: J
x , y ( x , y ) v v
x y

( u, v )
7. The functions u and v are called functionally dependent if 0.
( x, y)

( u, v ) ( x , y )
8. 1
( x , y ) ( u, v )
9. Taylors Expansion:

f ( x , y ) f (a , b)
1
1!

hf x (a , b) kf y (a , b)
1 2
2!

h f xx (a , b) 2hkf xy (a , b) k 2 f yy (a , b)

1 3
3!

h f xxx (a , b) 3h2 kf xxy (a , b) 3hk 2 f xyy (a , b) k 3 f yyy (a , b) ...

where h x a and k y b

Unit V (Multiple Integrals)

b x
1. f ( x , y )dxdy x : a to b and y : o to x (Here the first integral is w.r.t. y)
a 0

b y
2. f ( x , y )dxdy x : 0 to y and y : a to b (Here the first integral is w.r.t. x)
a 0

3. Area dxdy (or) dydx

R R

x r cos
To change the polar coordinate y r sin
dxdy rdrd
4. Volume dxdydz (or) dzdydx
V V

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GENERAL:

dx x dx
1. sin 1 (or) sin 1 x
a x2
a
2
1 x 2

2. a x
dx
2 2
log x a 2 x 2 (or)
dx
1 x 2
log x 1 x 2
dx 1 x dx
3. a tan 1 (or) tan 1 x
2
x 2
a a 1 x2

x 2 a2 x
4. a 2 x 2 dx
2
a x 2 sin 1
2 a
/2 /2
n1 n 3 2
sin x dx cos n x dx if n is odd and n 3
n
5. . ... .1
0 0
n n2 3
/2 /2
n1 n 3 1
6.
0
sin n x dx cos n x dx
0
. ... .
n n2 2 2
if n is even

----All the Best----

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