Engineering Mathematics

Engineering Mathematics - I (Calculus & Linear Algebra)

1. Differential Calculus

• Limits and Continuity
• Differentiation of functions of one variable
• Mean Value Theorems: Rolle’s, Lagrange’s, and Cauchy’s
• Taylor’s and Maclaurin’s series
• Partial Derivatives, Total Derivatives
• Jacobians and Hessians
• Maxima and Minima for multivariable functions (Lagrange multipliers)

2. Integral Calculus

• Definite and Indefinite Integrals
• Techniques of Integration (Substitution, Partial Fractions, By Parts)
• Improper Integrals
• Beta and Gamma Functions
• Applications of Integrals: Area, Volume, Arc Length, Surface Area

3. Linear Algebra

• Systems of Linear Equations
• Matrix Operations
• Inverse and Rank of a Matrix
• Eigenvalues and Eigenvectors
• Diagonalization
• Cayley-Hamilton Theorem

Engineering Mathematics - II (Differential Equations & Complex Analysis)

1. Ordinary Differential Equations (ODEs)

• First-order ODEs (Separable, Exact, Linear, Bernoulli)
• Second and Higher-order Linear ODEs
• Method of Undetermined Coefficients
• Variation of Parameters
• Applications: Mechanical and Electrical Systems

2. Laplace Transforms

• Definition and Properties
• Inverse Laplace Transform
• Convolution Theorem
• Applications to ODEs

3. Complex Numbers and Complex Functions

• Analytic Functions
• Cauchy-Riemann Equations
• Conformal Mapping
• Complex Integration
• Cauchy’s Integral Theorem and Formula
• Laurent Series, Residues, and Residue Theorem

Engineering Mathematics - III (Vector Calculus, PDEs, and Transforms)

1. Vector Calculus

• Gradient, Divergence, and Curl
• Directional Derivatives
• Line, Surface, and Volume Integrals
• Green’s, Gauss’s, and Stokes’ Theorems (with applications)

2. Partial Differential Equations (PDEs)

• Formation and Classification of PDEs
• Solution of First-order Linear PDEs (Lagrange's Method)
• Method of Separation of Variables
• One-dimensional Heat and Wave Equations
• Laplace Equation

3. Fourier Series and Transforms

• Fourier Series: Dirichlet’s conditions, Half-range Expansions
• Fourier Integral and Fourier Transform
• Sine and Cosine Transforms
• Applications to Signal and Heat Problems

Engineering Mathematics - IV (Probability, Statistics & Numerical Methods)

1. Probability and Statistics

• Probability Theory and Axioms
• Conditional Probability and Bayes’ Theorem
• Random Variables and Probability Distributions
• Expectation, Variance, and Moments
• Binomial, Poisson, Normal Distributions
• Curve Fitting and Correlation
• Regression Analysis
• Hypothesis Testing

2. Numerical Methods

• Errors and Approximations
• Root-finding Methods: Bisection, Newton-Raphson
• Interpolation: Newton’s and Lagrange’s methods
• Numerical Differentiation and Integration (Trapezoidal, Simpson’s)
• Numerical Solution of ODEs: Euler, Runge-Kutta methods