(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 |

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:

- Monday 5/9, 13:30–15:30 (near
**Aldo's Café**, in the Discovery Building). - Wednesday 5/11, 14:00–15:30 (in my offce, Van Vleck 503).

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.

See the official syllabus.

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.

Math 222 and 234.

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% |

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 | [solutions] | |

Midterm exam 2 | ||

Final exam | Wednesday May 11, 2016 at 17:05–19:05, Ingraham 19 |

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 R^{n}; 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 |