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