Below are listed the approximate topics covered in class and recommended homework for Math 21a. The dates listed are for the Mon-Wed-Fri sections. Your section may differ slightly in topics, problems, and dates. The Tues-Thurs sections will cover the same topics each week, but the topics and the recommended problems will be divided differently. Weekly mandatory problem sets will generally be posted on Tuesday or Wednesday and will be due one week later.

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Homework and detailed syllabus from February

Problem Set #1
Solutions to Problem Set #1

Problem Set #2
Solutions to Problem Set #2

Problem Set #3 and Exam Study Guide
Solutions to Problem Set #3

Mar 2: (Sections 2.5 and 5.2) Remarks on continuity and differentiability. The gradient vector field of a function of several variables. Conservative vector fields, independence of path, and the Fundamental Theorem of Line Integrals. Case of F as the velocity field of a fluid and the line integral of F around a simple closed curve giving the "vorticity" of the fluid. Algebraic properties of the line integral. Explanation of why the line integral is (oriented) reparametrization invariant using the Chain Rule. Homework: 2.5/1 and 5.1/3d and 5.2/5,7cde,8.

Mar 5: (Sections 5.2 and 2.6) Finish up conservative vector fields and the Fundamental Theorem of Line Integrals. Higher order derivatives & quadratic approx. Equality of mixed partial derivatives (in most circumstances). Derivative test for when a vector field is conservative. Hessian matrix of 2^nd derivatives. Homework: 2.6/3

Mar 7: Review for Midterm Exam I.

Mar 7, 4pm to 5:30pm: First Midterm Exam (Science Center Hall C)

Mar 9: (Sections 2.6 and 2.7) Further discussion about the geometry of 2nd derivatives and quadratic approximation. Examples of partial differential equations in physics (optional). Homework: 2.6/5,7 and 2.7/1

Mar 12: (Section 2.7) Extrema of functions of several variables. Unconstrained optimization. Vanishing of the gradient at (differentiable) extreme points. Identification of different types of extreme points (minima, maxima, and saddle points) using the Hessian matrix and 2nd derivative test (when possible). Fact that extrema may occur at stationary points, points of nondifferentiability, and at points on the boundary of the domain.
Homework: pgs 2.7/3,7,9,19,21,23

Problem Set #4
Problem Set #4 solutions, Part 1
Problem Set #4 solutions, Part 2
Problem Set #4 solutions, Part 3
Problem Set #4 solutions, Part 4
Problem Set #4 solutions, Part 5

Mar 14: (Sections 2.7 & 4.4 and Lagrange Multipliers Supplement) Constrained optimization and the Method of Lagrange multipliers (Ñ f = l Ñ g) . Finding extrema on the boundary of a domain - by parametrization of the boundary or by Lagrange Multipliers. Observations regarding optimization in unbounded domains.  Homework: 2.7/11 and 4.4/1,3,5,7 and the following problems:
1) Find the minimum distance from the surface x² + y² - z² = 1 to the origin.
2) Find the maximum and minimum values of x + y - 2z on the sphere x² + y² + z² = 1.
3) Find the maximum and minimum of xyz on the surface x + y + 4z = 1.
4) Model the earth as the sphere x² + y² + z² = 1. Suppose that the temperature at a point (x, y, z) on the surface is T(x, y, z) = x² - y² + z + 1 in appropriate units. Find the points with the highest and lowest temperatures.
5) Suppose that the profit from Scooter sales is a function N of three variables (x, y, z) given by N(x, y, z) = - 4x² + 2xy - z². Suppose as well that the values of (x, y, z) are not independent, but constrained by x + y + 2z = 1. What values of (x, y, z) should be used to maximize the profit?

Mar 16: Applications in economics.
Homework: Exercises in the Lagrange Multipliers supplement.

Mar 19: General chain rule. Implicit differentiation. Economics applications.
Homework: Finish problem set #4.

Mar 21: Method of Least Squares. Finish up details of optimization, implicit diferentiation, etc.
Homework: to appear on Problem Set #5

Problem Set #5 (due no later than Thursday after Spring Break)

Mar 23: (Sections 3.1 and 3.2) Integration over regions in R² and R³. Average value of a function. [Some sections will begin this topic after Spring Break.] Homework: work on problem set.

Mar 24 - Apr 1: Spring Break

Question Center: In addition to class, problem sessions, and office hours, the Mathematics Department operates the Math Question Center in Loker on Sunday, Monday, Tuesday, Wednesday, and Thursday evenings from 8pm to 10pm. The Question Center will be staffed by Course Assistants from Math 1a, 1b, 21a, and 21b and by graduate students and others. You are encouraged to use this resource as you do your homework and when questions arise. It is intended to supplement the office hours held by your Section Leader.