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.

### Syllabus

### Taught by

Steve Brunton