About Course
Engineering Mathematics is designed for students with little math backgrounds to learn Applied Mathematics in the most simple and effective way. The aim of this course is to provide students with the knowledge of not only mathematical theories but also their realworld applications so students understand how and when to use them.
This Engineering Mathematics course is divided into 2 sections. Section 1 (the first 28 lectures) covers the most fundamental math that anyone should learn of ME564. Here you will learn everything about ME564. At the end of section 1, you should have a better understanding of the ME564 classes. Section 2 (29 Lecture ) introduces you to the world of calculus. Here you will learn the fundamental definition of integration and differentiation. You will also learn the most commonlyused rules and properties through simple examples and full knowledge of the ME565 class lecture.
At the end of this course, you should have a good understanding of all the topics covered in this course and be able to use them in realworld applications.
Who this Engineering Mathematics course is for:
 Students who wish to learn applied mathematics. That is, how mathematics is used in the practical world.
 Students who are going to learn more advanced engineering courses. Most intermediate engineering courses will require the basic math covered in this course.
Course Content
Engineering Mathematics

ME564 Lecture 1: Overview of engineering mathematics
41:16 
ME564 Lecture 2: Review of calculus and first order linear ODEs
48:43 
ME564 Lecture 3: Taylor series and solutions to first and second order linear ODEs
53:23 
ME564 Lecture 4: Second order harmonic oscillator, characteristic equation, ode45 in Matlab
00:00 
ME564 Lecture 5: Higherorder ODEs, characteristic equation, matrix systems of first order ODEs
00:00 
ME564 Lecture 6: Matrix systems of first order equations using eigenvectors and eigenvalues
00:00 
ME564 Lecture 7: Eigenvalues, eigenvectors, and dynamical systems
00:00 
ME564 Lecture 8: 2×2 systems of ODEs (with eigenvalues and eigenvectors), phase portraits
00:00 
ME564 Lecture 9: Linearization of nonlinear ODEs, 2×2 systems, phase portraits
00:00 
ME564 Lecture 10: Examples of nonlinear systems: particle in a potential well
00:00 
ME564 Lecture 11: Degenerate systems of equations and nonnormal energy growth
00:00 
ME564 Lecture 12: ODEs with external forcing (inhomogeneous ODEs)
00:00 
ME564 Lecture 13: ODEs with external forcing (inhomogeneous ODEs) and the convolution integral
00:00 
ME564 Lecture 14: Numerical differentiation using finite difference
00:00 
ME564 Lecture 15: Numerical differentiation and numerical integration
00:00 
ME564 Lecture 16: Numerical integration and numerical solutions to ODEs
00:00 
ME564 Lecture 17: Numerical solutions to ODEs (Forward and Backward Euler)
00:00 
ME564 Lecture 18: RungeKutta integration of ODEs and the Lorenz equation
00:00 
ME564 Lecture 19: Vectorized integration and the Lorenz equation
00:00 
ME564 Lecture 20: Chaos in ODEs (Lorenz and the double pendulum)
00:00 
ME564 Lecture 21: Linear algebra in 2D and 3D: inner product, norm of a vector, and cross product
00:00 
ME564 Lecture 22: Div, Grad, and Curl
00:00 
ME564 Lecture 23: Gauss’s Divergence Theorem
00:00 
ME564 Lecture 24: Directional derivative, continuity equation, and examples of vector fields
00:00 
ME564 Lecture 25: Stokes’ theorem and conservative vector fields
00:00 
ME564 Lecture 26: Potential flow and Laplace’s equation
00:00 
ME564 Lecture 27: Potential flow, stream functions, and examples
00:00 
ME564 Lecture 28: ODE for particle trajectories in a timevarying vector field
00:00 
ME565 Lecture 1: Complex numbers and functions
00:00 
ME565 Lecture 2: Roots of unity, branch cuts, analytic functions, and the CauchyRiemann conditions
00:00 
ME565 Lecture 3: Integration in the complex plane (CauchyGoursat Integral Theorem)
00:00 
ME565 Lecture 4: Cauchy Integral Formula
00:00 
ME565 Lecture 5: ML Bounds and examples of complex integration
00:00 
ME565 Lecture 6: Inverse Laplace Transform and the Bromwich Integral
00:00 
ME565 Lecture 7: Canonical Linear PDEs: Wave equation, Heat equation, and Laplace’s equation
00:00 
ME565 Lecture 8: Heat Equation: derivation and equilibrium solution in 1D (i.e., Laplace’s equation)
00:00 
ME565 Lecture 9: Heat Equation in 2D and 3D. 2D Laplace Equation (on rectangle)
00:00 
ME565 Lecture 10: Analytic Solution to Laplace’s Equation in 2D (on rectangle)
00:00 
ME565 Lecture 11: Numerical Solution to Laplace’s Equation in Matlab. Intro to Fourier Series
00:00 
ME565 Lecture 12: Fourier Series
00:00 
ME565 Lecture 13: Infinite Dimensional Function Spaces and Fourier Series
00:00 
ME565 Lecture 14: Fourier Transforms
00:00 
ME565 Lecture 15: Properties of Fourier Transforms and Examples
00:00 
ME565 Lecture 16: Discrete Fourier Transforms (DFT)
00:00 
ME565 Lecture 16 Bonus: DFT in Matlab
00:00 
ME565 Lecture 17: Fast Fourier Transforms (FFT) and Audio
00:00 
ME565 Lecture 18: FFT and Image Compression
00:00 
ME565 Lecture 19: Fourier Transform to Solve PDEs: 1D Heat Equation on Infinite Domain
00:00 
ME565 Lecture 20: Numerical Solutions to PDEs Using FFT
00:00 
ME565 Lecture 21: The Laplace Transform
00:00 
ME565 Lecture 22: Laplace Transform and ODEs
00:00 
ME565 Lecture 23: Laplace Transform and ODEs with Forcing and Transfer Functions
00:00 
ME565 Lecture 24: Convolution integrals, impulse and step responses
00:00 
ME565 Lecture 25: Laplace transform solutions to PDEs
00:00 
ME565 Lecture 26: Solving PDEs in Matlab using FFT
00:00 
ME 565 Lecture 27: SVD Part 1
00:00 
ME565 Lecture 28: SVD Part 2
00:00 
ME565 Lecture 29: SVD Part 3
00:00 
The Laplace Transform: A Generalized Fourier Transform
00:00