## Math 211: Multivariable Calculus Study Guide for Exam 2

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. While this exam will emphasize the topics below, some of these chapters do rely on older material, so it is important that you remember 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).

14.3: You must know how to find the partial derivatives of a several variable function, including higher order partial derivatives. Clairaut's Theorem is especially helpful for these higher order derivatives.

14.4: You should know what it means for a several variable function to be differentiable in terms of being well approximated by a tangent plane. (You do not need to know Definition 7 on p. 942, but you should know the result in Theorem 8 below.) You should be able to find the equation for the tangent plane to the graph as well as the linearization and linear approximation of a function at a given point.

14.5: You should be able to carry out the chain rule, in general. You should also be able to use the chain rule for differentiating implicit functions.

14.6: Know how to find the gradient of a function and the directional derivative in the direction of a given vector. You should also know their geometric interpretations, including the fact that the gradient points in the direction of maximum rate of change and is normal to level curves/surfaces. Know how the gradient can be used to find the tangent plane to surface defined implicitly.

14.7: Know how to identify and classify critical points of a function, and how to find global maxima and minima on a closed and bounded domain.

14.8: Be able to use Lagrange multipliers to optimize a function subject to one constraint equation. (You do not need to know how to use Lagrange multipliers with several constraints.) Note that the methods in this section can also be used to find the maximum and minimum values of a function on the boundary of a closed, bounded domain.

15.1-15.3: These sections cover double integrals over rectangles and more general domains in ℝ2. Make sure you know how to find the limits of integration and how to find the double integral by calculating the iterated integral. Also know how to switch limits of integration, and be able to identify situations in which the integral can only be solved analytically by switching these limits. You should also know how to obtain the area of a given domain using a double integral.

15.4: You should know how to set up a double integral over a domain best described in polar coordinates. In particular, you should be able to identify situations in which using polar coordinates is useful, know how to find the limits of integration, and how to convert the integrand to a function in polar coordinates.

15.7: This section is analogous to 15.1-15.3. You should once again be able to find limits of integration, and to find the triple integral by calculating the iterated integral. Once again, you should be able to switch limits of integration. You should also know how to find the volume of a solid, and its mass and center of mass, using a double integral.

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.

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
14.3 5-8, 15-44, 53-58, 63-72
14.4 1-6, 11-21
14.5 1-16, 21-36, 38-41
14.6 4-17, 21-26, 31-34, 40-46, 53-55
14.7 1-18, 29-36, 39-51
14.8 3-14, 19-21
14 R 13-17, 19-22, 25-29, 32, 33, 35-37, 42-48, 51-56, 59-61
15.2 3-22, 25-31, 35, 36
15.3 1-10, 17-32, 43-56
15.4 1-27, 29-32, 39
15.7 3-22, 29-36, 39-42
15.8 1-12, 17-30
15.9 1-15, 19-36, 39-41
15 R 3-34, 41, 42, 47, 48