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
**Trajectory**

A curve which cuts every member of a given family of curves in accordance with some given law, is called trajectory of the given family of curves.

**Orthogonal Trajectory**

If a cuts every member of given family of curves at right angle it is called orthogonal trajectory.

As an example y = mx and are respectively the family of straight line and family of circles with centre or origin and radius a.

Every line y = mx cut each member of family of circles at right angle.

So y = mx is orthogonal trajectory of

**Determination of orthogonal trajectories in cartesiancoordinates**

**Working Rule.**

1. Differentiate the given equation of family of curves. Eliminate the parameters between the derived equation and given equation of the family. It will give the differential equation of given family of curve.

2. Replace dy/dx (slope of tangent) by –dx/dy (slope of orthogonal family) in the above differential equation in step 1.

3. Obtain the general solution of differential equation in step 2.

**Self orthogonal family of Curves**

If each member of a given family of curves intersects all other members orthogonally, then given family of curves is said to be self orthogonal.

Let the equation of given family of curves be f(x, y, c) = 0 …(1)

Differentiate (1) w.r.t x and eliminate c between (1) and derived equation

We shall arrive at the differential equation of given family as F(x, y, dy/dx) = 0 …(2)

Let be the angle, which the tangent at P to the member PQ with x axis.

Then …(3)

Let (X, Y) be the current coordinates of any point of a trajectory. At point of intersection P of any members of (2) with the trajectory , let be the angle which the tangent to the trajectory makes with x axis.

So, …(4)

Let PT and PT’ intersect at 90°.

So

At the point of intersection x = X and y = Y

So , is the required family of trajectories.

**Determination of orthogonal trajectories in polarcoordinate system**

Let the equation of given family of curves be in polar form …(1)

Where c is a parameter

Differentiate (1) w.r.t and eliminating c between (1) and derived equation. We get the differential

equation of the given family of curves.

Let be that differential equation …(2)

Let be the angle between the tangent PT to a member PQ of the given family of curves and radius vector OP at any point .

then …(3)

Let be the current coordinates of any point of a trajectory. At point of intersection P of any member of (2) with the trajectory PQ’, let be the angle which the tangent PT’ to the trajectory makes with the common radius vector OP.

Then

Let PT and PT’ intersect at 90°

Then

So using (3) and (4)

At the point of intersection r = r’ and

So is the required orthogonal trajectory.

**Determination of Orthogonal trajectories in PolarCoordinates**

**Working Rule :**

1. Differentiate the given equation of family of curves w.r.t (generally take logarithm). Eliminate the parameter.

2. Replace by and obtain the differential equation of orthogonal trajectories

3. Obtain the general solution of differential equation obtained above.

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