Differential Calculus and Linear Algebra

Matrix Algebra: Introduction to Matrices, Elementary Transformation, Inverse by E- transformation, Rank of a Matrix by Echelon Form, Solution of system of linear Equations by direct method (Gauss Elimination Method), and Iterative method (Gauss Seidel Method), Eigen Values & Eigen Vectors, Diagonalization, Applications : Machine Design, Finite Element Analysis, Dynamics of Mechanical Systems.

Introduction of Limits, continuity and differentiability for function of two variables, Partial derivatives, Composite functions, Total derivative, Euler’s Theorem for homogeneous functions, Taylor’s and Maclaurin’s expansion for function of two variables, Jacobians, properties of Jacobian (without proof), Maxima & Minima for a function of up to two variables. Applications: Optimization and Machine Design, Fluid Mechanics, Control Systems, Stress and Strain Analysis.

Evaluation of double and triple integrals, change of order of integration, Change of variable (double -integral).Beta and Gamma functions, Properties of Beta and Gamma function. Dirichlet’s Integral. Application of double integrals and triple integral to find area and volume. Applications: Area of Irregular Surfaces, Volume of Solids, Fluid Mechanics, Heat Transfer.

Vector spaces (over the field of real numbers), Subspaces, Spanning set, Linear independence, Basis and Dimension. Linear transformations, Range and Null space, Rank- Nullity theorem, Matrix of a linear transformation. Applications: Force & Moment Analysis, Mechanics of Machines, Mechanical Vibrations, CAD.

Scalar point function, Vector point function, Gradient of a scalar field, Directional derivatives, Divergence and curl of a vector function and their application to solenoidal and irrotational vectors respectively .Integration of Vectors, Stokes Theorem and Gauss Divergence Theorem (Without Proof) . Applications: Fluid flow analysis, Aerodynamics, Drone navigation systems.