**Mathematical Methods for Physical Sciences I**

**Math 20000 Section 80**

**Fall 2017**

First day handout (updated version, tentative outline included)

**Announcements**

- **The University Registrar changed the final exam schedules**, see "General Information" below for the new date.

- There will be NO problem session on Mon, Oct 30th, but I'll be in my office from 5pm to 7pm if you have questions.

- An e-mail was sent out in the afternoon of Oct 26 containing some information about the midterm.

**General Information**

** **Instructor: Da Rong (Daren) Cheng

Office: Eckhart 409

Office hours: Tue 9:30 - 10:50am, Thu 2:00 - 3:20pm

E-mail: chengdr "at" uchicago.edu

Class hours: TTh 3:30 - 4:50pm - Eckhart 202

Textbook: *Vector Calculus (6th ed.)*, Marsden & Tromba (Chapters 1 - 8)

Problem sessions: M 6:00 - 7:00pm - Eckhart 117

Exam dates:

In-class midterm on Tuesday, Oct 31st [PDF][Solutions]

Final exam on Tuesday, Dec 5th, 4:00 - 6:00pm (**Note change of date.**)

Grading:

Homework 10%

Midterm 40%

Final 50%

**Homework **

Homework will generally be assigned every Thursday and due the following Thursday by the end of class. Late homework will not be accepted under all circumstances. However, at the end of the quarter, the lowest homework score will be dropped. This means that you can miss one homework without penalty.

Homework 1 (due 10/5) [PDF][Solutions]

Homework 2 (due 10/12) [PDF][Solutions]

Homework 3 (due 10/19) [PDF][Solutions]

Homework 4 (due 10/26) [PDF][Solutions]

Homework 5 (due 11/2) [PDF][Solutions]

Homework 6 (due 11/9) [PDF][Solutions]

Homework 7 (due 11/16) [PDF][Solutions]

Homework 8 (due **11/28**) [PDF][Solutions]

**Topics covered **

This part will be updated twice a week to record the topics actually covered in each lecture. The numbers in parentheses refer to the sections in Marsden & Tromba to which the topics roughly belong.

**Week 1**

9/26 Operations on vectors; length, distance, the innerproduct (dot product) and angle; area and the determinant of 2 by 2 matrices. (1.1 - 1.3)

9/28 Area and the cross product; equations for lines and planes; curves, paths and tangent lines. (1.1 - 1.3; 2.4)

**Week 2**

10/3 Graphs and level sets (2.1); the method of sections (2.1); tangent planes to graphs (2.3, p.110) and level sets (2.6, p.138-139).

10/5 Tangent planes and affine approximations (2.3); first special case of the chain rule (2.5); second order partial derivatives (3.1).

(The picture I drew after computing the affine approximation of f(x, y) = x^2 - y^2 at (x, y) = (3, 1) is obviously incorrect. Here is a more accurate picture.)

**Week 3**

10/10 Second order Taylor approximations (3.2); local (relative) and global (absolute) extrema; the first-derivative test for local extrema (3.3).

10/12 The second derivative test for local extrema (3.3).

** Week 4**

10/17 Constrained extremum and the method of Lagrange multipliers (3.4); global extrema on bounded domains (3.3).

10/19 The double integral over rectangles and elementary regions; reduction of double integrals to iterated integrals; applications to computing volumes. (5.1 - 5.3)

** Week 5**

10/24 More examples/non-examples of x-simple or y-simple regions; applications to computing volumes (continued). (5.1 - 5.3)

10/26 Volumes of regions other than tetrahedrons; polar coordinates; derivative of vector-valued functions.

** Week 6**

10/31 Midterm.

11/2 Derivative of vector-valued functions (continued) (2.3); matrix multiplication (1.5); second special case of the chain rule (2.5); the divergence (4.4).

** Week 7**

11/7 The curl (4.4); length of paths (4.2); the path integral (integral of real-valued functions along paths) (7.1).

11/9 Parametrized surfaces (to be continued) (7.3); Area of surfaces (7.4); double integral in polar coordinates (6.2, p.321).

** Week 8**

11/14 Surface integral of real-valued functions (7.5); the line integral (integral of vector-valued functions along paths) (7.2).

11/16 The line integral of gradients; The line integral along curves with prescribed orientation.

**Problem session material **

This part contains the worksheets we go over in problem sessions. I do not plan to write up solutions to each of them, but you're of course welcome to discuss any problem with me.

Worksheet 1 (10/9) [PDF]

Worksheet 2 (10/16) [PDF]

Worksheet 3 (10/23) [PDF]