Overview

This course on Engineering Mathematics aims to provide a comprehensive overview of mathematical concepts and tools essential for graduate studies in Mechanical Engineering. The course covers topics such as calculus, differential equations, linear algebra, complex numbers, Fourier analysis, Laplace transforms, and numerical methods. Students will learn to apply these mathematical techniques to solve engineering problems, analyze dynamical systems, and understand concepts like potential flow and vector fields. The teaching method includes lectures, examples, and numerical exercises. This course is intended for graduate students in Mechanical Engineering or related fields who want to strengthen their mathematical foundation for engineering applications.

Syllabus

ME564 Lecture 1: Overview of engineering mathematics.
ME564 Lecture 2: Review of calculus and first order linear ODEs.
ME564 Lecture 3: Taylor series and solutions to first and second order linear ODEs.
ME564 Lecture 4: Second order harmonic oscillator, characteristic equation, ode45 in Matlab.
ME564 Lecture 5: Higher-order ODEs, characteristic equation, matrix systems of first order ODEs.
ME564 Lecture 6: Matrix systems of first order equations using eigenvectors and eigenvalues.
ME564 Lecture 7: Eigenvalues, eigenvectors, and dynamical systems.
ME564 Lecture 8: 2x2 systems of ODEs (with eigenvalues and eigenvectors), phase portraits.
ME564 Lecture 9: Linearization of nonlinear ODEs, 2x2 systems, phase portraits.
ME564 Lecture 10: Examples of nonlinear systems: particle in a potential well.
ME564 Lecture 11: Degenerate systems of equations and non-normal energy growth.
ME564 Lecture 12: ODEs with external forcing (inhomogeneous ODEs).
ME564 Lecture 13: ODEs with external forcing (inhomogeneous ODEs) and the convolution integral.
ME564 Lecture 14: Numerical differentiation using finite difference.
ME564 Lecture 15: Numerical differentiation and numerical integration.
ME564 Lecture 16: Numerical integration and numerical solutions to ODEs.
ME564 Lecture 17: Numerical solutions to ODEs (Forward and Backward Euler).
ME564 Lecture 18: Runge-Kutta integration of ODEs and the Lorenz equation.
ME564 Lecture 19: Vectorized integration and the Lorenz equation.
ME564 Lecture 20: Chaos in ODEs (Lorenz and the double pendulum).
ME564 Lecture 21: Linear algebra in 2D and 3D: inner product, norm of a vector, and cross product.
ME564 Lecture 22: Div, Grad, and Curl.
ME564 Lecture 23: Gauss's Divergence Theorem.
ME564 Lecture 24: Directional derivative, continuity equation, and examples of vector fields.
ME564 Lecture 25: Stokes' theorem and conservative vector fields.
ME564 Lecture 26: Potential flow and Laplace's equation.
ME564 Lecture 27: Potential flow, stream functions, and examples.
ME564 Lecture 28: ODE for particle trajectories in a time-varying vector field.
ME565 Lecture 1: Complex numbers and functions.
ME565 Lecture 2: Roots of unity, branch cuts, analytic functions, and the Cauchy-Riemann conditions.
ME565 Lecture 3: Integration in the complex plane (Cauchy-Goursat Integral Theorem).
ME565 Lecture 4: Cauchy Integral Formula.
ME565 Lecture 5: ML Bounds and examples of complex integration.
ME565 Lecture 6: Inverse Laplace Transform and the Bromwich Integral.
ME565 Lecture 7: Canonical Linear PDEs: Wave equation, Heat equation, and Laplace's equation.
ME565 Lecture 8: Heat Equation: derivation and equilibrium solution in 1D (i.e., Laplace's equation).
ME565 Lecture 9: Heat Equation in 2D and 3D. 2D Laplace Equation (on rectangle).
ME565 Lecture 10: Analytic Solution to Laplace's Equation in 2D (on rectangle).
ME565 Lecture 11: Numerical Solution to Laplace's Equation in Matlab. Intro to Fourier Series.
ME565 Lecture 12: Fourier Series.
ME565 Lecture 13: Infinite Dimensional Function Spaces and Fourier Series.
ME565 Lecture 14: Fourier Transforms.
ME565 Lecture 15: Properties of Fourier Transforms and Examples.
ME565 Lecture 16 Bonus: DFT in Matlab.
ME565 Lecture 17: Fast Fourier Transforms (FFT) and Audio.
ME565 Lecture 16: Discrete Fourier Transforms (DFT).
ME565 Lecture 18: FFT and Image Compression.
ME565 Lecture 19: Fourier Transform to Solve PDEs: 1D Heat Equation on Infinite Domain.
ME565 Lecture 20: Numerical Solutions to PDEs Using FFT.
ME565 Lecture 21: The Laplace Transform.
ME565 Lecture 22: Laplace Transform and ODEs.
ME565 Lecture 23: Laplace Transform and ODEs with Forcing and Transfer Functions.
ME565 Lecture 24: Convolution integrals, impulse and step responses.
ME565 Lecture 25: Laplace transform solutions to PDEs.
ME565 Lecture 26: Solving PDEs in Matlab using FFT.
ME 565 Lecture 27: SVD Part 1.
ME565 Lecture 28: SVD Part 2.
ME565 Lecture 29: SVD Part 3.
The Laplace Transform: A Generalized Fourier Transform.