**Unit – 1: Differential Calculus – I**

Leibnitz’s Theorem

Partial Derivatives

Euler’s Theorem for Homogeneous Functions

Total Derivatives

Change of Variables

Curve Tracing: Cartesian and Polar Coordinates

**Unit – 2: Differential Calculus – II**

Taylor’s And McLaurin’s Theorems

Expansion of Function of Several Variables

Jacobean

Approximation of Errors

Extrema of Functions of Several Variables

Lagrange’s Method of Multipliers (Simple Applications)

**Unit – 3: Linear Algebra**

Inverse of a Matrix by Elementary Transformations

Rank of a Matrix (Echelon & Normal Form)

Linear Dependence

Consistency of Linear System of Equations and Their Solution

Characteristic Equation

Eigen Values and Eigen Vectors

Cayley-Hamilton Theorem

A Brief Introduction to Vector Spaces, Subspaces

Rank & Nullity

Linear Transformations

**Unit – 4: Multiple Integrals**

Double and Triple Integrals

Change of Order of Integration

Change of Variables

Application of Integration to Lengths

Volumes and Surface Areas – Cartesian and Polar Coordinates

Beta and Gamma Functions

Dirichlet’s Integral and Applications

**Unit – 5: Vector Calculus**

Point Function

Gradient

Divergence and Curl and Their Physical Interpretations

Vector Identities

Directional Derivatives

Line, Surface and Volume Integrals

Applications of Green’s, Stoke’s And Gauss Divergence Theorems (Without Proofs)