Calculus and Linear Algebra

Introduction of Limits, continuity and differentiability for function of two variables, Partial derivatives, Composite functions, Total derivative, Euler’s Theorem for homogeneous functions, Change of variables Expansion for function of two variables, Jacobians, properties of Jacobian (without proof), Maxima & Minima for a function of several variables, Lagrange’s method of multipliers. Approximation of Errors. Applications: AI model optimization, Cloud computing performance, Data processing systems, Machine learning optimization, Neural network tuning, Resource allocation problem.

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: Image processing, Data density estimation, AI probability distribution.

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: AI heat mapping, Robotics path planning, Computer graphics vector fields.

Introduction to Matrices, Complex Matrices: Hermitian, Skew Hermitian and Unitary Matrix. 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, Similarity Transformation, Diagonalization, Stability Interpretation. Applications: Error detection and correction codes, Computer Graphics Rendering , Loop Current Analysis, Principal Component Analysis (PCA), Face recognition systems.

Definition of Field, Introduction to Vector Spaces, Basic Properties of Vector Spaces, Sub spaces, Linear dependence and independence of vectors, Basis and Dimension. Application: Engineering Data Modeling and Compression, Signal representation and processing, Noise reduction using vector space concepts .