Differential 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, 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. Application: Determining reaction rates, mass transfer and diffusion studies, optimization of chemical process.

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: Determining mass and energy distribution, calculation of reactor and storage tank volume.

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, Application: Process simulation and optimization, chemical reaction network analysis.

Linear differential equation of nth order with constant coefficients, Cauchy- Euler equation, solution of second order linear differential equations with variable coefficient by the method of variation of parameters, change of independent variable, normal form. Application: Chemical reaction Kinetics, reactor design, mass transfer.

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 in pipes and reactors, computational fluid dynamics (CFD).