Math 321: Applied Mathematical Analysis I (Spring 2016) (Vector and Complex Calculus for the Physical Sciences)

 Lecture Room: 104 Van Hise Lecture Time: 13:20–14:10 MWF Lecturer: Jean-Luc Thiffeault Office: 503 Van Vleck Email: Office Hours: Mon 2:10–3:00, Wed 11:50–12:30

Final Exam

Don't forget that the final is on Wednesday May 11, 2016 at 17:05–19:05 in Ingraham 19.

The final exam covers the material of Chapters 2 and 3. There will be a definite emphasis on complex variables and complex integration. Update: The exam will consist of 5 questions, 3 of which deal with complex variables (60% of marks).

Here are some things that you should definitely know how to do for the final:

• Take div, grad, curl of a function.
• Show some basic identities involving div, grad, curl.
• Know the basic integral theorems (Green, Stokes, Gauss, ...)
• Know how to do line, surface, and volume integrals.
• Know how to compute the flux of a vector field through a surface.
• Know the basic properties of complex numbers and how to take logs, roots, etc.
• Know how to integrate in the complex plane following a path.
• Know Cauchy's theorem and when to apply it.
• Be able to derive Cauchy's formula and its general form from Cauchy's theorem.
• Know how to evaluate an integral over a closed contour by finding the residues of a function and/or by applying Cauchy's formula.
• Be able to use residue calculus to evaluate real integrals by choosing an appropriate contour in the complex plane, and showing how the original real integral relates to the complex integral.

Office hours on exam week:

Ivan's review sessions on exam week:

• Monday 5/9, 12:00–13:30, Ingraham 120.
• Tuesday 5/10, 12:00–13:30, Ingraham 120.

Syllabus

See the official syllabus.

Textbook

There is no official textbook for the class. We will follow Prof. Waleffe's lecture notes fairly closely.

Here are some textbooks you might consult for some extra exercises and background. Be warned that they may use different notation and present the material in a different order from the class.

• Div, Grad, Curl, and All That: An Informal Text on Vector Calculus by H. M. Schey.
• Vector Calculus by J. E. Marsden and A. Tromba.
• Schaum's Outline of Vector Analysis by S. Lipschutz and M. R. Spiegel.

Prerequisites

Math 222 and 234.

Homework

Every week a few questions will be assigned, mostly from the course notes. The homework will be due on Friday in class. One or two random questions will be explictly graded, and the rest graded for completion.

The homework is posted on the webpage of Ivan Ongay Valverde, the TA for the class.

There will be two midterm exams and a cumulative final exam. The final grade will be computed according to:

 Homework 20% Midterm exam 1 20% Midterm exam 2 20% Final exam 40%

Exam Dates

The midterm exams will be given in the evening on the dates below. No notes, calculators, or anything else other than basic writing implements is allowed.

 Midterm exam 1 Monday February 22, 2016 at 17:30–19:00, room B130 [solutions] Midterm exam 2 Monday April 4, 2016 at 17:30–19:00, room B130 Final exam Wednesday May 11, 2016 at 17:05–19:05, Ingraham 19

Schedule of Topics

 lecture date pages topic 1 01/20 – Introduction; Autonomous flight (TED talk) 2 01/22 1–5 Review of vectors in 2D and 3D 3 01/25 6–10 Addition and scaling of vectors; General vector spaces 4 01/27 11–13 Points and coordinates 5 01/29 12–19 Dot product; Orthonormal bases 6 02/01 20–24 Dot product and norm in Rn; Cross product; Double cross product 7 02/03 24–30 Orientation of bases; Levi–Civita symbol 8 02/05 30–34 Einstein sum convention 9 02/08 34–37 Mixed product and determinant 10 02/10 39–45 Lines and planes; Change of Cartesian basis 11 02/12 – Discussion of exercises 12 02/15 46–48 Matrices 13 02/17 48–52 Euler angles; Gram–Schmidt procedure 14 02/19 57–59 Vector calculus; Motion of a particle 15 02/22 – Discussion of exercises 16 02/24 59–62 Central force motion 17 02/26 66–69 Curves 18 02/29 66–70 Frenet–Serret formulas 19 03/02 70–73 Integrals along curves 20 03/04 73–79 Surfaces 21 03/07 80–81 Surface integrals 22 03/09 80–81 Examples of surface integrals 23 03/11 82–84 Volume integrals 24 03/14 85–87 Mappings and curvilinear coordinates 25 03/16 88,92 Change of variables 26 03/18 93–97 Gradient 27 03/28 97–99 Div and curl; vector identities 28 03/30 99–102 Div, grad, curl in polar coordinates 29 04/01 104–110 Fundamental theorems of vector calculus 30 04/04 – Discussion of exercises 31 04/06 110–116 Gauss' theorem 32 04/08 110–116 Gauss' theorem with singularities 33 04/11 117–122 Review of complex numbers 34 04/13 122–136 Review of complex numbers (cont'd) 35 04/15 137–141 The Cauchy–Riemann equations 36 04/18 151–152 Complex integration 37 04/20 152–154 Cauchy's theorem 38 04/22 154–157 Laurent series and residues 39 04/25 157–161 Cauchy's formula 40 04/27 161–163 Complex methods for real integrals 41 04/29 163–166 Complex methods for real integrals (cont'd) 42 05/02 168–169 Branch cuts 43 05/04 – Discussion of exercises 44 05/06 – Discussion of exercises