DOUBLE INTEGRAL

Change of Order

7 Topics
Transformation of Variables

8 Topics
BETA AND GAMMA FUNCTION

VOLUME

SURFACE AREA

Surface Area

4 Topics
GRADIENT, DIVERGENCE AND CURL

Divergence and Curl

4 Topics
LINE INTEGRAL

Line Integral

4 Topics
GREENS THEOREM

Greens Theorem

6 Topics
SURFACE INTEGRAL

Surface Integral

6 Topics
GAUSS DIVERGENCE THEOREM

Gauss Divergence Theorem

8 Topics
STOKES THEOREM

Stokes Theorem

5 Topics
CONSERVATIVE VECTOR FIELD

Conservative Vector Field

3 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 :**

If determine the value of where *C* is the curve in the *xy* plane from (0, 0) to (1, 2).

**Solution :**

The curve lies in *xy* plane, so,* z* = 0. *z* can never be taken as independent variable *z* is a dependent variable. Now, out of *x* and *y*, any one variable can be taken as independent.

Suppose *x* is taken as independent variable

So, the line integral reduces to a definite integral.

If *y* is taken as independent variable then* x* can be expressed in terms of *y* as

So,

So, the line integral reduces to a definite integral

**Example : 1 **

Evaluate around a circle

**Solution :**

Let *C* denotes the circle. The parametric equations of circle is

Here, *x* and *y* have been expressed in terms of parameter which varies from 0 to as one traverses the circle.

So,

Here, *r* is a constant, because integral is carried over a circle.

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