DOUBLE INTEGRAL

Change of Order

7 Topics
Transformation of Variables

8 Topics
BETA AND GAMMA FUNCTION

VOLUME

SURFACE AREA

Surface Area

4 Topics
LIMITS

Introduction to Limits

4 Topics
Methods of Finding Limits

10 Topics
CONTINUITY

Continuity

4 Topics
DIFFERENTIABILITY

Differentiability

4 Topics
APPLICATION OF DERIVATIVES

Monotonicity

5 Topics
Critical Points

3 Topics
Maxima and Minima

3 Topics
MEAN VALUE THEOREM

Lagrange Mean Value Theorem

3 Topics
Part B: FUNCTION OF TWO VARIABLES

Function of Two Variables

7 Topics
INTRODUCTION

DIFFERENTIAL EQUATIONS OF FIRST ORDER AND FIRST DEGREE

ORTHOGONAL TRAJECTORY

Orthogonal Trajectory

2 Topics
DIFFERENTIAL EQUATION WITH CONSTANT COEFFICIENTS

CAUCHY EULER EQUATIONS

DIFFERENTIAL EQUATION OF SECOND ORDER

Section 5: LINEAR ALGEBRA

Introduction to Matrices

7 Topics
Linear Equations

6 Topics
VECTOR SPACE AND LINEAR EQUATIONS

Revision of Matrices

6 Topics
Basis and Dimensions

5 Topics
ORTHOGONALITY

Orthogonality

6 Topics
EIGENVALUES AND EIGENVECTORS

Similar Matrices

1 Topic
LINEAR TRANSFORMATION

Matrix of Linear Transformation

2 Topics
Section 6: GROUP THEORY

Cayley Table

1 Topic
Unit Group

1 Topic
Group of Matrices

1 Topic
Dihedral Groups

1 Topic
Problems on Groups

1 Topic
CYCLIC GROUP

Cyclic Groups

3 Topics
PERMUTATION GROUP

Permutation Group

5 Topics
ISOMORPHISM OF GROUPS

Isomorphism of Groups

8 Topics
EXTRNAL DIRECT PRODUCT

External Direct Product

3 Topics
HOMOMORPHISM OF GROUP

Homomorphism of Groups

8 Topics
SET THEORY

REAL NUMBERS

Real Numbers

4 Topics
TOPOLOGY ON REAL LINE

Open Sets

4 Topics
Closed Sets

1 Topic
REAL SEQUENCES

Introduction to Real Sequences

4 Topics
Limit Points

3 Topics
Subsequence

1 Topic
Cauchy Sequence

1 Topic
INFINITE SERIES

Introduction to Infinite Series

2 Topics
Alternating Series

1 Topic
POWER SERIES

Power Series

1 Topic
**Example 1 :**

Find the orthogonal trajectories of family of circles , where g is the parameter.

**Solution :**

The given family of curves

…(1)

Differentiate both sides w.r.t. x,

…(2)

Put the value of g in (1)

…(3)

which is the differential equation of given family.

Now replace by in (3)

So

put

put in (4) …(5)

which is linear differential equation

Now

Multiply both sides (5) with IF, it becomes.

where d is the parameter.

**Example 2 :**

Show that one parameter family of curves are self orthogonal.

**Solution :**

We are given …(1)

Differentiate …(2)

Put value of c in (1)

(3) gives the differential equation of given family

Now replace y’ by in (3), we get the differential equation of orthogonal trajectory.

Which is same as in equation (3)

So, differential equation of given curve and differential equation of its orthogonal trajectaries are same. So, the family of curves is self orthogonal.

**Example 3 :**

Find the orthogonal trajectories of the family of curves , where is a parameter.

**Solution :**

We are given

Differentiate both sides w.r.t x

…(1)

On solving

Now on substituting values of and in (i)

which is the differential equation is given family. …(2)

Now replace with to obtain differential equation of orthogonal trajectory

Differential equation (2) and (3) are same, which gives the differential equation of family. It’s orthogonal trajectories are same. So the family of curves are self orthogonal.

**Example 4 :**

Show that the families of curves given by the equation and intersect orthogonally.

**Solution :**

Here we have to show that the family of orthogonal trajectory of the family of curves

We have …(1)

On taking logrithm both sides

Differentiate both sides w.r.t

…(2)

which is free from parameter. So, it is the differential equation of given family.

Now replace with in (2)

On integrating

which is the required orthogonal trajectory.

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