## Math 211: Multivariable Calculus Study Guide for the Final Exam

The final exam is on Tuesday, December 20, from 2:00pm to 5:00pm, in Beneski 107.

Reading/exam period office hours are F 1:00pm-3:00pm and M 3:00-5:00pm.

The Q-Center will continue to hold afternoon and evening Calculus drop-in hours, but please be sure to consult the updated schedule.

The following is a chapter by chapter guide intended to help you organize the material we have covered in class as you study for your exam. It is only intended to serve as a guideline, and may not explicitly mention everything that you need to study.

The final exam is cumulative, so it is important that you study the material from earlier chapters as well. It is also still important you are comfortable with the basics of differentiation and integration (as covered in Calculus I and Calculus II classes).

15.8 & 15.9: You should know how to convert between the various co-ordinate systems (ie rectangular, cylindrical and spherical), including how to describe sets of points or functions given in one system in terms of one of the others. You should also know how to convert integrals in rectangular co-ordinates to integrals in one of the other (polar) co-ordinate systems, both in terms of finding the new limits, and knowing the correct expansion factor by which you need to multiply the integrand. Practising problems here will help you become more comfortable with identifying the most efficient co-ordinate system to use for a given problem.

15.10: Know how to find the Jacobian determinant and the change of variables formula for double integrals. (It is also helpful to be able to calculate the Jacobian determinant in 3 dimensions for the change of variables to polar coordinates.)

16.1: This section deals with the basics of vector fields, and what they represent. It is really important to understand this section since all subsequent sections build on this material.

16.2 This section generalizes arc length integrals from Section 13.3. You should be able to find the line integral of a scalar function and of a vector field over a given space curve. You should also be able to interpret the vector line integrals physically and geometrically.

16.3 You must know the Fundamental Theorem for Line Integrals, and how to identify when a given vector field is conservative. You should also know how to find a potential function for such vector fields, and be able to identify situations where this is useful for evaluating an integral.

16.4:You must know the statement of Green's Theorem, and be able to use it to obtain the value of a given integral. (You should be able to see how this gives us 2-dimensional analogue to the Fundamental Theorem of Calculus.) This theorem is especially useful for finding the line integral of a (possibly non-conservative) vector field by treating the curve over which you are integrating as part of a boundary of a region in the plane. Make sure you know how to identify the boundary of a given region, and how to determine the appropriate orientation of the boundary.

16.5: You should also be comfortable with finding the curl and divergence of vector fields and know how to use the ideas of curl and divergence to determine whether a given vector field has a scalar or vector potential.

16.6: Make sure you practice finding parametrizations of surfaces, especially for cones, cylinders and spheres, along with their parametrization domains (ie, the range of values in which the parameters vary) and the normal vectors for the surface. Know how to use integrals to find the surface area of given surface.

16.7: You should also know how to find the surface integral of a scalar function or a vector field over a given surface, and how to interpret this physically and geometrically. Note that the surface integral of a vector field can be related to that of a scalar function in an analogous way to that for line integrals.

16.8-16.9: It will be helpful to know the statement of Stokes's Theorem and the Divergence Theorem and how they are used in computing integrals. In particular, you should be able to see how these theorems give analogues to the Fundamental Theorem of Calculus (along with Green's Theorem and the Fundamental Theorem of Line Integrals). Make sure you know how to identify the boundary of a given surface or volume, and how to determine its orientation.

Please be sure to review homework and example problems for the chapters given above! Below is a list of relevant additional practice problems for each section. An "R" in the section number refers to the review exercises at the end of each chapter.

Section Suggested Problems
15.8 1-12, 17-30
15.9 1-15, 19-36, 39-41
15.10 1-6, 11-17
15 R 3-34, 41, 42, 47-49
16.1 1-7, 11-18, 21-26, 29-32
16.2 1-22, 39-42, 47, 49
16.3 1-20, 30
16.4 1-14, 17, 18
16.5 1-31, 38
16.6 1, 2, 13-26, 33-36, 39-50
16.7 5-32
16 R 1-21, 25, 26

The departmental list of comprehensive exam topics may also provide a useful summary of things to review.

Old Math 211 finals are available here.