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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Apr 2: (Sections 3.1 and 3.2) Integration over regions in R² and R³. Iterated integrals and the Fubini Theorem. Average value of a function. Using integration to calculate area, volume, mass, population, etc.
Homework: Read sections 3.1 and 3.2 and Appendix B. Do problems 3.1/7,8,9,11 and 3.2/2,3,5,6,8.

Apr 4: (Section 3.3, 3.4 & Triple Integrals Supplement) Integrals in polar coordinates (R²) and cylindrical coordinates (R³).
Homework: 3.2/11 and 3.3/2,3,5,7 and 3.4/1abcde,3,5.

Apr 6: (Section 3.4 and Center of Mass supplement) Triple integrals in cylindrical and spherical coordinates. Examples of centroids and averaging. Homework: 3.4/1f,2,4,7,9,11,15.

Problem Set #6 and Midterm Exam #2 Study Guide (due no later than Fri, April 13)
Problem Set #6 Solutions

Apr 9: (Section 3.5) Weighted averages, center of mass, other applications of multiple integrals. Change of variables in multiple integrals.

Apr 11: Review for Midterm Exam #2

Apr 11: Midterm Exam #2 - 4:00pm - 5:30pm in Sci Ctr Hall C

Apr 13: (Section 3.5) Change of Variables Theorem for Multiple Integrals. Invented coordinates for regions in R² and R³. Interpretation and use of Jacobian determinants. Homework: 3.5/5,7,9,10,11.

Apr 16: (Section 5.3) Finish up last details of change of variables. Review of Fundamental Theorem for Line Integrals. Green's Theorem.
Homework: 5.3/1,3,7,9

Problem Set #7 (due no later than Tues, April 24)
Problem Set #7 Solutions

Apr 18: (Sections 5.3 -5.4) Examples using Green's Theorem (for regions with or without holes). Parametrization of surfaces and introduction to surface integrals. Homework: 5.3/11 and 5.4/1,2,3,4

Apr 20: (Section 5.5, 5.6 and Surface Area Supplement) Surfaces integrals; surface area; flux of a vector field through a surface.
Homework: 5.5/1,2,3,5,6

Apr 23: (Section 5.6) Flux of a vector field through a surface. Further calculations of surface integrals. Algebraic definitions of the divergence and curl of a vector field in R³. Homework: 5.6/1,3,5,7,9.

Problem Set 8 (due no later than Thurs, May 3)
Problem Set #8 Solutions

Apr 25: (Section 5.6 and 5.7) Geometric definition of the divergence of a vector field. Statement and proof of the Divergence Theorem. Homework: 5.7/3,5,6,9,10

Apr 27: (Section 5.7) Derive algebraic definition of the div F from the geometric definition. Geometric definition of the curl of a vector field. Statement and proof of Stokes' Theorem.  Homework: 5.7/1,7,8,11

Apr 30: (Section 5.7 and class notes) Recap of Stokes' Theorem. Show that the geometric definition of the curl of a vector field implies the algebraic definition of curl F.  Prove Green's Theorem as a corollary of Stokes' Theorem.

Problem Set 9 (due no later than Tues, May 15)
Problem Set #9 Solutions

May 2: Examples and applications of the Divergence Theorem and Stokes' Theorem. Five versions of the Fundamental Theorem of Calculus. Applications in the derivation of partial differential equations in physics.

May 4: Last details, more examples and applications.

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.