Principles of Applied Mathematics

Lecture Summaries

LEC #	TOPICS	LECTURE SUMMARIES
1	Overview of Class	Lecture 1 Summary (PDF)
2	General Framework. Discrete to Continuum Limits.	Lecture 2 Summary (PDF)
3	Sources for Conservation Laws	Lecture 3 Summary (PDF)
4	Conservation Laws in 2-D or 3-D, Dimensional Analysis, Fick’s Law	Lecture 4 Summary (PDF)
5	Conservation Laws and Partial Differential Equations (PDEs), Method of Characteristics, Traffic Flow	Lecture 5 Summary (PDF)
6	Linearized Traffic Flow	Lecture 6 Summary (PDF)
7	Characteristic Curves, Nonlinear Problems	Lecture 7 Summary (PDF)
8	Nonlinear Kinematic Waves and Wave-Breaking	Lecture 8 Summary (PDF)
9	Traffic Flow Examples	Lecture 9 Summary (PDF)
10	Shock Jump Condition	Lecture 10 Summary (PDF)
11	Domain of Dependence, Characteristics, Shocks	Lecture 11 Summary (PDF)
12	Flood Waves	Lecture 12 Summary (PDF)
13	Ordinary Differential Equations (ODEs), Shock and Rarefaction	Lecture 13 Summary (PDF)
14	Arrow of Time	Lecture 14 Summary (PDF)
15	Extended Theory	Lecture 15 Summary (PDF)
16	EIKO, Waves and Caustics	Lecture 16 Summary (PDF)
17	Whole Curves, Envelopes, Mach Cone	Lecture 17 Summary (PDF)
18	Domain of Dependence, 1st Order PDEs	Lecture 18 Summary (PDF)
19–20	Review of Theory: Expansion Fans, Shocks, Entropy Increase at Shocks	Lecture 19–20 Summary (PDF)
21	Entropy, Quadratic Flow	Lecture 21 Summary (PDF)
22	Envelopes and Relationship Between Conservation Laws and Shocks	Lecture 22 Summary (PDF)
23	Shock Structure Due to Higher Order Effects	Lecture 23 Summary (PDF)
24	Gas Dynamics, Acoustics, and Strings	Lecture 24 Summary (PDF)
25	Gas Dynamics, Polytropic Gas, Hyperbolic	Lecture 25 Summary (PDF)
26	Linear Algebra for Gas Dynamics	Lecture 26 Summary (PDF)
27	Linear Gas Dynamics: Acoustic in a Pipe	Lecture 27 Summary (PDF)
28	Gauss / Stokes Theorem	Lecture 28 Summary (PDF)
29	General Solution for the Wave Equation	Lecture 29 Summary (PDF)
30	Multivariable Calculus, Differentiability, Properties of the Gradient, Taylor Expansion, Implicit Function Theorem	Lecture 30 Summary (PDF)
31	Characteristics for Klein-Gordon	Lecture 31 Summary (PDF)
32	Characteristics, Simple Waves, and Shocks for Gas Dynamics, Numerical Method by Characteristics	Lecture 32 Summary (PDF)
33	Separation of Variables and Normal Modes	Lecture 33 Summary (PDF)
34	Scalar Product Definition of Self-Adjoint	Lecture 34 Summary (PDF)
35	Generalize to Operators, Finite Differences Schemes for PDEs	Lecture 35 Summary (PDF)
36–39	FD Schemes, Introduction to the vNSA Problem Series, Artificial Viscosity, von Neumann Stability Analysis for Numerical Schemes	Lecture 36–39 Summary (PDF)