Algebra, Vector Analysis And Geometry (Mathematics) Paper-I

Units

Topics

I

1.1 Historical background:

1.1.1 Development of Indian Mathematics Later Classical Period (500-1250)

1.1.2 A brief biography of Varahmihira and Aryabhatta

1.2 Rank of a Matrix

1.3 Echelon and Normal form of a matrix

1.4 Characteristic equations of a matrix

1.4.1 Eigen-Value

1.4.2 Eigen-Vectors

II

2.1 Cayley Hamilton theorem.

2.2 Application of Cayley Hamilton theorem to find the inverse of a matrix.

2.3Application of matrix to solve a system of linear equations.

2.4 Theorems on consistency and inconsistency of a system of linear equations.

2.5 Solving linear equations up to three unknowns.

III

3.1 Scalar and Vector products of three and four vectors.

3.2 Reciprocal vectors

3.3 Vector differentiation

3.3.1 Rules of differentiation

3.3.2 Derivatives of Triple Products

3.4 Gradient, Divergence and Curl

3.5 Directional derivatives

3.6 Vector Identities

3.7 Vector Equations

IV

4.1 Vector Integration

4.2 Gauss Theorem (without proof) and problems based on it

4.3 Green theorem (without proof) and problems based on it

4.4 Stoke theorem (without proof) and problems based on it

V

5.1 General equation of second degree

5.2 Tracing of conics

5.3 System of conics

5.4 Cone

5.4.1 Equation of cone with given base

5.4.2 Generators of cone

5.4.3 Condition for three mutually perpendicular generators

5.4.4 Right circular cone

5.5 Cylinder

5.5.1 Equation of cylinder and its properties

5.5.2 Right Circular Cylinder

5.5.3 Enveloping Cylinder