Engineering Mathematics

Polar forms, De Moivre's theorem, and the Cauchy-Riemann equations.

• Complex Numbers and Analytic Functions

  “Polar forms, De Moivre's theorem, and the Cauchy-Riemann equations.”

• Complex Integration

  “Contour integrals and Cauchy’s Integral Theorem.”

• Taylor and Laurent Series

  “Power series expansions in the complex plane and singularity classification.”

• Residue Theorem

  “Evaluating real integrals using complex residue calculus.”

• Conformal Mapping

  “Transforming complex domains for fluid flow and heat transfer applications.”

Representing periodic functions as sums of sines and cosines.

• Fourier Series and Harmonic Analysis

  “Representing periodic functions as sums of sines and cosines.”

• Fourier Transforms

  “Continuous transforms for non-periodic signals and spectral analysis.”

• Separation of Variables

  “Fundamental technique for solving linear boundary value problems.”

• The Heat Equation

  “Modeling thermal diffusion in one and two dimensions.”

• The Wave Equation

  “Analyzing vibration and wave propagation in strings and membranes.”

• Laplace's Equation

  “Solving steady-state potential problems in various coordinate systems.”

Linearity, shifting theorems, and transforms of derivatives.

• Definition and Properties of Laplace Transforms

  “Linearity, shifting theorems, and transforms of derivatives.”

• Inverse Laplace Transforms

  “Partial fraction decomposition and lookup table applications.”

• Solving IVPs with Laplace Transforms

  “Transforming differential equations into algebraic equations for solution.”

• Step and Impulse Functions

  “Modeling discontinuous forcing functions using Heaviside and Dirac Delta functions.”

• Convolution Integrals

  “Applying the convolution theorem to find system responses.”

• Transfer Functions and Stability

  “Analyzing system behavior in the s-domain and pole-zero mapping.”

Solving consistent and inconsistent systems using Gaussian elimination and Echelon forms.

• Systems of Linear Equations and Row Reduction

  “Solving consistent and inconsistent systems using Gaussian elimination and Echelon forms.”

• Matrix Algebra and Invertibility

  “Operations, properties of inverses, and the Invertible Matrix Theorem.”

• Determinants and Cramer's Rule

  “Calculation of determinants and their application in solving linear systems.”

• Vector Spaces and Subspaces

  “Analyzing basis, dimension, and the null/column spaces of matrices.”

• Eigenvalues and Eigenvectors

  “Characteristic equations, diagonalization, and similarity transformations.”

• Orthogonality and Least Squares

  “Inner products, Gram-Schmidt process, and normal equations for overdetermined systems.”

Bisection, Secant, and Newton-Raphson methods.

• Root-Finding Algorithms

  “Bisection, Secant, and Newton-Raphson methods.”

• Numerical Integration and Differentiation

  “Trapezoidal rule, Simpson’s rules, and finite difference approximations.”

• Numerical Solutions to ODEs

  “Euler’s method and Runge-Kutta (RK4) techniques.”

• Interpolation and Curve Fitting

  “Lagrange polynomials, splines, and regression analysis.”

• Finite Difference Methods for PDEs

  “Discretizing partial differential equations for computational solvers.”

Separable, linear, exact, and Bernoulli equations with initial value problems.

• First-Order Differential Equations

  “Separable, linear, exact, and Bernoulli equations with initial value problems.”

• Higher-Order Linear Homogeneous Equations

  “Constant coefficient equations and the Wronskian for linear independence.”

• Method of Undetermined Coefficients

  “Solving non-homogeneous equations with polynomial, exponential, and trig forcing functions.”

• Variation of Parameters

  “General method for finding particular solutions to non-homogeneous ODEs.”

• Systems of Linear Differential Equations

  “Coupled ODEs solved via eigenvalue methods and matrix exponentials.”

• Power Series Solutions

  “Solving equations near ordinary and singular points using Frobenius method.”

Binomial, Poisson, Normal, and Exponential distributions in engineering.

• Probability Distributions

  “Binomial, Poisson, Normal, and Exponential distributions in engineering.”

• Joint and Marginal Distributions

  “Handling multiple random variables and covariance analysis.”

• Point and Interval Estimation

  “Confidence intervals and maximum likelihood estimation.”

• Hypothesis Testing

  “Z-tests, T-tests, and P-value interpretation for quality control.”

• Linear Regression and Correlation

  “Modeling relationships between variables and least-squares fitting.”

• Reliability and Risk Analysis

  “Failure rate modeling and system reliability calculations.”

Multivariable differentiation and total derivatives in engineering contexts.

• Partial Derivatives and Chain Rules

  “Multivariable differentiation and total derivatives in engineering contexts.”

• Gradient, Divergence, and Curl

  “Vector differential operators and their physical interpretations in fluid and EM fields.”

• Line Integrals and Work

  “Integration along curves and the Fundamental Theorem for Line Integrals.”

• Surface Integrals and Flux

  “Calculating flow through surfaces and parameterized surface integration.”

• Green's and Stokes' Theorems

  “Relating line integrals to surface and area integrals in vector fields.”

• The Divergence Theorem

  “Applications of Gauss's Theorem to volume integrals and flux conservation.”