Subject
Chapters/Topics
Brief Description
Linear Algebra
1. Vectors and Vector Spaces
Introduces vectors in ℝⁿ, linear independence, basis, dimension, subspaces, and inner product spaces.
2. Matrices and Determinants
Covers matrix operations, rank, inverse, determinants, properties, and applications in solving linear systems.
3. Systems of Linear Equations
Gaussian elimination, Gauss-Jordan method, LU decomposition, consistency, and solution techniques.
4. Eigenvalues and Eigenvectors
Characteristic equation, diagonalization, symmetric matrices, quadratic forms, and spectral theorem.
5. Linear Transformations
Kernel, image, matrix representation, change of basis, and similarity transformations.
Calculus
6. Differential Calculus
Limits, continuity, derivatives, chain rule, implicit differentiation, higher-order derivatives, and applications (maxima/minima, curve sketching).
7. Integral Calculus
Riemann integration, fundamental theorems, techniques (substitution, parts, partial fractions), improper integrals, and applications (area, volume).
8. Multivariable Calculus
Partial derivatives, gradient, directional derivatives, chain rule, double/triple integrals, line/surface integrals, and Jacobian.
9. Vector Calculus
Vector fields, divergence, curl, Green’s, Stokes’, and Gauss’ divergence theorems with engineering applications.
Differential Equations
10. First-Order ODEs
Separable, exact, linear, Bernoulli, homogeneous equations, and applications (growth/decay, mixing problems).
11. Higher-Order Linear ODEs
Homogeneous and non-homogeneous equations, characteristic equation, method of undetermined coefficients, variation of parameters, and Cauchy-Euler equations.
12. Systems of ODEs
Matrix method, eigenvalues approach, phase plane analysis, and stability.
13. Laplace Transforms
Definition, properties, inverse transforms, convolution, solving ODEs with initial conditions, and discontinuous forcing functions.
14. Partial Differential Equations
Classification, separation of variables, heat, wave, and Laplace equations with boundary conditions.
Complex Analysis
15. Complex Numbers and Functions
Algebra of complex numbers, polar form, analytic functions, Cauchy-Riemann equations, and harmonic functions.
16. Complex Integration
Contour integrals, Cauchy’s integral theorem and formula, residues, and residue theorem for real integrals.
Fourier Series and Transforms
17. Fourier Series
Periodic functions, orthogonal functions, even/odd extensions, half-range series, and Gibbs phenomenon.
18. Fourier Transforms
Definition, properties, convolution theorem, Parseval’s relation, and applications in signal processing.
Numerical Methods
19. Error Analysis and Interpolation
Round-off/truncation errors, Lagrange and Newton interpolation, finite differences, and spline interpolation.
20. Numerical Solutions of Equations
Bisection, Newton-Raphson, secant, fixed-point iteration, and convergence criteria.
21. Numerical Differentiation and Integration
Finite difference formulas, trapezoidal, Simpson’s rules, Romberg integration, and Gaussian quadrature
22. Numerical Solutions of ODEs
Euler, Runge-Kutta (2nd & 4th order), predictor-corrector methods, and stability analysis.
23. Finite Difference Methods for PDEs
Explicit/implicit schemes for parabolic, hyperbolic, and elliptic PDEs; consistency, stability, and convergence.
24. Probability Theory
Probability and Statistics
Axioms, conditional probability, Bayes’ theorem, random variables, expectation, and variance.
25. Probability Distributions
Discrete (binomial, Poisson) and continuous (normal, exponential, gamma) distributions, moment-generating functions
26. Statistical Inference
Sampling distributions, point estimation, confidence intervals, hypothesis testing, chi-square, t, and F tests.
27. Regression and Correlation
Linear regression, least squares, multiple regression, correlation coefficient, and ANOVA.
Discrete Mathematics
28. Set Theory and Logic
Sets, relations, functions, propositional logic, predicates, and proof techniques.
29. Graph Theory
Graphs, paths, cycles, trees, connectivity, Eulerian/Hamiltonian graphs, shortest path, and minimum spanning trees.