XE-A Engineering Mathematics

Matrix Fundamentals: Determinants, matrix inversion, and rank determination.

Systems of Equations: Solvability conditions for unique, infinite, or no solutions.

Spectral Analysis: Eigenvalues and eigenvectors (symmetric matrices), matrix diagonalization, and Cayley-Hamilton Theorem.

Single Variable Calculus: Limits, L'Hospital's rule, continuity, mean value theorems, Taylor's theorem, improper integrals, and geometric applications (areas/volumes of revolution).

Multivariable Calculus: Partial and directional derivatives, total derivatives, saddle points, Lagrange multipliers, and double integrals.

Sequences & Series: Convergence tests (ratio, root, integral), power series, and Fourier series for $2\pi$-periodic functions.

Vector Fields: Gradient, divergence, curl, line integrals, and Green's theorem.

Complex Analysis: Argand plane, polar forms, De Moivre's theorem, analytic functions, and Cauchy-Riemann equations.

Ordinary Differential Equations (ODEs): First-order linear/nonlinear equations, higher-order linear ODEs with constant/variable coefficients, Cauchy-Euler equations, Wronskian, variation of parameters, and series solutions.

Partial Differential Equations (PDEs): Linear second-order PDE classification and separation of variables for the 1D heat and 2D Laplace equations.

Probability & Statistics: Axioms, conditional probability, Bayes' theorem, discrete/continuous distributions (Binomial, Poisson, Normal), and linear regression.

Numerical Techniques: Linear systems (LU decomposition, Gauss elimination), root finding (Newton-Raphson), interpolation (Lagrange, Newton), integration (Trapezoidal, Simpson's rules), and explicit Euler's method for ODEs.