Math 4340 (Advanced Engineering Mathematics), Summer II 2017 (online)

Math 4340 (Advanced Engineering Mathematics), Summer II 2017 (online)

"Mathematics, rightly viewed, possesses not only truth, but supreme beauty." --Bertrand Russell

About the class

This course is an introduction to Fourier Series and Partial Differential Equations. Throughout the country, these topics are taught in a variety of contexts -- from a very theoretical course on PDEs and Applied Analysis for senior math majors, to a more computational course geared torwards engineers, e.g., a "Differential Equations II" class. My goal in this course is to strike a balance between these two extremes. I have included enough basic linear algebra (vector spaces, independence, basis, inner products, self-adjoint operators) so the students can see the mathematical structure behind the scenes. However, I have omitted advanced details such as Hilbert spaces, and different types of norms and convergence (pointwise, uniform, and in norm). My goal is for this to be useful to math, science, and engineering majors alike.

Class essentials



Links to the individual lectures are listed below. Or, you can view the full YouTube playlist here. To avoid blurriness, these are best viewed by changing the settings to 720p (High Definition) rather than the default of 240p. This can be easily done by clicking the "wheel" on the lower right corner; right next to the "cc" button.

Editor's note: Any YouTube link with (??:??) means I haven't made the video yet.

Section 1: Some linear algebra. (4 lectures: 2 hr 55 min).

Section 2: Linear differential equations. (7 lectures: 4 hr 51 min). Section 3: Fourier series. (8 lectures: 5 hr 16 min.) Section 4: Boundary value problems and Sturm-Liouville theory. (6 lectures: 4 hr 1 min). Section 5: Partial differential equations (PDEs) on bounded domains. (4 lectures, 2 hrs 47 min)--> Section 6: PDEs on unbounded domains. Section 7: Higher-dimensional PDEs. (5 lectures: 3 hr 57 min).


Homework should be written up carefully and concisely. Please write in complete sentences. Part of your grade will be based on the presentation and clarity of your answers.

HW 1: pdf | tex. Topics: Vector spaces, linear independence, and bases. Due Friday, June 30, 2017.
HW 2: pdf | tex. Topics: Linear maps, inner products, and orthogonality. Due Monday, July 3, 2017.
HW 3: pdf | tex. Topics: The Wronskian, affine spaces, and linear ODEs. Due Thursday, July 6, 2017.
HW 4: pdf | tex. Topics: Cauchy-Euler equations and power series solutions. Due Monday, July 10, 2017.
HW 5: pdf | tex. Topics: The Frobenius method and Bessel's equation. Due Thursday, July 13, 2017.
HW 6: pdf | tex. Topics: Real Fourier series, and Fourier sine & cosine series. Due Monday, July 17, 2017.
HW 7: pdf | tex. Topics: Complex Fourier series, Fourier transforms, and Parseval's theorem. Due Wednesday, July 19, 2017.
HW 8: pdf | tex. Topics: Self-adjoint operators and Sturm-Liouville theory. Due Friday, July 21, 2017.
HW 9: pdf | tex. Topics: Diffusion and the heat equation. Due Monday, July 24, 2017.
HW 10: pdf | tex. Topics: The transport, wave, and Schrödinger equation. Due Thursday, July 27, 2017.
HW 11: pdf | tex. Topics: Harmonic functions and higher dimensional PDEs. Due Monday, July 31, 2017.
HW 12: pdf | tex. Topics: PDEs in other coordinate systems. Due Thursday, August 3, 2017.