An open-source textbook covering vector calculus, ordinary and partial differential equations, and Fourier series. The textbook is used in a first-year graduate level course in the Department of Mechanical Engineering at the Colorado School of Mines. It undergoes extensive revisions annually, but is relatively complete.

This course provides the fundamental computational toolbox for solving science and engineering problems. Topics include review of linear algebra, applications to networks, structures, estimation, finite difference and finite element solutions of differential equations, Laplace's equation and potential flow, boundary-value problems, Fourier series, the discrete Fourier transform, and convolution. We will also explore many topics in AI and machine learning throughout the course.

This course provides a review of linear algebra, including applications to networks, structures, and estimation, Lagrange multipliers. Also covered are: differential equations of equilibrium; Laplace's equation and potential flow; boundary-value problems; minimum principles and calculus of variations; Fourier series; discrete Fourier transform; convolution; and applications.
Note: This course was previously called "Mathematical Methods for Engineers I."

Differential Equations are the language in which the laws of nature are expressed. Understanding properties of solutions of differential equations is fundamental to much of contemporary science and engineering. Ordinary differential equations (ODE's) deal with functions of one variable, which can often be thought of as time.

The laws of nature are expressed as differential equations. Scientists and engineers must know how to model the world in terms of differential equations, and how to solve those equations and interpret the solutions. This course focuses on the equations and techniques most useful in science and engineering.
Course Format
This course has been designed for independent study. It provides everything you will need to understand the concepts covered in the course. The materials include:

Lecture Videos by Professor Arthur Mattuck.
Course Notes on every topic.
Practice Problems with Solutions.
Problem Solving Videos taught by experienced MIT Recitation Instructors.
Problem Sets to do on your own with Solutions to check your answers against when you're done.
A selection of Interactive Java® Demonstrations called Mathlets to illustrate key concepts.
A full set of Exams with Solutions, including practice exams to help you prepare.

Content Development
Haynes Miller 
Jeremy Orloff 
Dr. John Lewis 
Arthur Mattuck

This is a free textbook which covers material for an introductory course on differential equations with some partial differential equations material, though it assumes knowledge of matrix theory. It includes a section on computing Fourier series of polynomials. It also includes a link to the freely available student solutions manual.

The first term in a Fourier series is the average value (DC value) of the function being approximated. Created by Sal Khan.

This course continues the content covered in 18.100 Analysis I. Roughly half of the subject is devoted to the theory of the Lebesgue integral with applications to probability, and the other half to Fourier series and Fourier integrals.

Learn how to make waves of all different shapes by adding up sines or cosines. Make waves in space and time and measure their wavelengths and periods. See how changing the amplitudes of different harmonics changes the waves. Compare different mathematical expressions for your waves.

Learn how to make waves of all different shapes by adding up sines or cosines. Make waves in space and time and measure their wavelengths and periods. See how changing the amplitudes of different harmonics changes the waves. Compare different mathematical expressions for your waves.

This course covers the same material as Differential Equations (18.03) with more emphasis on theory. In addition, it treats mathematical aspects of ordinary differential equations such as existence theorems.

This is a undergraduate course. It will cover normed spaces, completeness, functionals, Hahn-Banach theorem, duality, operators; Lebesgue measure, measurable functions, integrability, completeness of L-p spaces; Hilbert space; compact, Hilbert-Schmidt and trace class operators; as well as spectral theorem.

This course introduces quantitative approaches to understanding brain and cognitive functions. Topics include mathematical description of neurons, the response of neurons to sensory stimuli, simple neuronal networks, statistical inference and decision making. It also covers foundational quantitative tools of data analysis in neuroscience: correlation, convolution, spectral analysis, principal components analysis, and mathematical concepts including simple differential equations and linear algebra.

Learn Differential Equations: Up Close with _Gilbert Strang and_ Cleve Moler is an in-depth series of videos about differential equations and the MATLAB® ODE suite. These videos are suitable for students and life-long learners to enjoy.
About the Instructors
Gilbert Strang is the MathWorks Professor of Mathematics at MIT. His research focuses on mathematical analysis, linear algebra and PDEs. He has written textbooks on linear algebra, computational science, finite elements, wavelets, GPS, and calculus.
Cleve Moler is chief mathematician, chairman, and cofounder of MathWorks. He was a professor of math and computer science for almost 20 years at the University of Michigan, Stanford University, and the University of New Mexico.
These videos were produced by The MathWorks and are also available on The MathWorks website.

This course covers the classical partial differential equations of applied mathematics: diffusion, Laplace/Poisson, and wave equations. It also includes methods and tools for solving these PDEs, such as separation of variables, Fourier series and transforms, eigenvalue problems, and Green's functions.

This course provides students with the basic analytical and computational tools of linear partial differential equations (PDEs) for practical applications in science engineering, including heat / diffusion, wave, and Poisson equations. Analytics emphasize the viewpoint of linear algebra and the analogy with finite matrix problems. Numerics focus on finite-difference and finite-element techniques to reduce PDEs to matrix problems. The Julia Language (a free, open-source environment) is introduced and used in homework for simple examples.