Multivariable Calculus Workbook Developed by: Jerry Morris, Sonoma State University A Companion to Multivariable Calculus McCallum, Hughes-Hallett, et. al. c Wiley, 2007 Note to Students: (Please Read) This workbook contains examples and exercises that will be referred to regularly during class. Please purchase or print out the rest of the workbook before our next class and bring it to class with you every day. 1. To Purchase the Workbook. Go to the Sonoma State University campus bookstore, where the workbook is available for purchase. The copying charge will probably be between $10.00 and $20.00. 2. To Print Out the Workbook. Go to the Canvas page for our course and click on the link \Math 261 Workbook", which will open the file containing the workbook as a .pdf file. BE FOREWARNED THAT THERE ARE LOTS OF PICTURES AND MATH FONTS IN THE WORKBOOK, SO SOME PRINTERS MAY NOT ACCURATELY PRINT PORTIONS OF THE WORKBOOK. If you do choose to try to print it, please leave yourself enough time to purchase the workbook before our next class in case your printing attempt is unsuccessful. 2 Sonoma State University Table of Contents Course Introduction and Expectations .............................................................3 Preliminary Review Problems ........................................................................4 Chapter 12 { Functions of Several Variables Section 12.1-12.3 { Functions, Graphs, and Contours . 5 Section 12.4 { Linear Functions (Planes) . 11 Section 12.5 { Functions of Three Variables . 14 Chapter 13 { Vectors Section 13.1 & 13.2 { Vectors . 18 Section 13.3 & 13.4 { Dot and Cross Products . 21 Chapter 14 { Derivatives of Multivariable Functions Section 14.1 & 14.2 { Partial Derivatives . 29 Section 14.3 { Local Linearity and the Differential . 34 Section 14.4 { Gradients and Directional Derivatives in the Plane . 38 Section 14.5 { Gradients and Directional Derivatives in Space . 44 Section 14.6 { The Chain Rule . 48 Section 14.7 { Second-Order Partial Derivatives . 51 Chapter 16 { Integration Section 16.1 { The Definite Integral of a Function of Two Variables . 54 Section 16.2 { Iterated Integrals . 60 Section 16.3 { Triple Integrals . 65 Section 16.4 { Integrals in Polar Coordinates . 72 Section 16.5 { Integration in Cylindrical and Spherical Coordinates . 78 Chapter 17 { Parameterized Curves Section 17.1 { Parameterized Curves . 86 Section 17.2 { Motion, Velocity, and Accleration . 94 Section 17.3 { Vector Fields . 98 Chapter 18 { Line Integrals Sections 18.1 & 18.2 { Line Integrals . 101 Section 18.3 { Gradient Vector Fields and Path Independent Fields . 109 Section 18.4 { Green's Theorem . 119 Chapter 19 { Flux Integrals Section 19.1 & 19.2 { Flux Integrals . 123 Chapter 20 { Calculus of Vector Fields Section 20.1 { The Divergence of a Vector Field . 139 Section 20.2 { The Divergence Theorem . 145 Section 20.3 { The Curl of a Vector Field . 151 Section 20.4 { Stokes' Theorem . 158 Multivariable Calculus Workbook, Wiley, 2007 3 Multivariable Calculus { An Introduction Welcome to Math 261, the world of calculus in 3 dimensions! In this course, we take many of the concepts from first-semester calculus, such as derivatives and integrals, and extend them to higher dimensions. Since the focus and expectations in this course are somewhat different than those of previous calculus courses you've taken, I want to take a few minutes to specifically point some of these out. 1. Unlike calculus 2 and parts of calculus 1, this course is mainly concept-driven rather than procedure-driven. In other words, understanding the practical and geometric significance of a calculation is often just as important as knowing how to do the calculation. For example, knowing how to calculate a derivative is nice, but understanding that it represents a rate of change is more useful in the real world. Therefore, there will be a lot problems in this class where you will need to choose for yourself an appropriate calculation method based on your understanding of the concept behind the method. This ability should develop naturally over the course of the semester provided that you take seriously the \give an explanation" portion of homework problems. In most cases, simply memorizing how to do a calculation but having no understanding of its significance will not get you very far. So don't be afraid to ask me those \why" types of questions early and often both inside and outside of class; the more curious you are about the deeper meaning behind what we do, the easier things will be for you! 2. Unlike some other calculus textbooks, this textbook is intended to be read by students both in the worked out examples and in the explanations between the examples. When you are stuck on a homework problem and are looking for assistance in the textbook, you won't always find a worked-out example exactly like it; rather, you may have to read the section and try to gain a deeper understanding of the big picture before you can attempt the problem. While I understand that this can be frustrating and time consuming, the goal is for you to gain a deeper understanding of the concepts in the course, and this understanding cannot be obtained by just copying the steps of similar worked-out examples. Keep in mind that making a sincere effort to read the book is important to your success in the course. Also keep in mind that you are not expected to understand everything in the textbook on your own! So don't be shy about asking for help when you are stuck, this is a normal part of the course that I am happy to help you with. 3. Due to the amount of material and the time that it takes to cover it, more is expected of you outside of class in multivariable calculus than in previous calculus courses. Since a typical problem in this course tends to take a long time, there will not be time to do as many different types of examples together in class as we could in a previous calculus course. Therefore, it will be important for you to carefully study examples in the textbook, to keep up with the list of practice problems I ask you to work through, and to be diligent about asking for help when you need it. This outside-of-class involvement in the course will be critical for you to be successful in the course. 4. One thing you will immediately notice about the exercises in our textbook is that a lot of them call for detailed explanations. While some students tend to write off such questions as being \not that important, since the calculations are what really matter in a math class", I couldn't disagree more! Mathematical calculations are of no use if their significance and connection to the real world cannot be explained. Therefore, I will expect you to show work when you do calculations and to give complete explanations when you are asked to explain or describe something. In particular, here are some specific grading criteria for homework, quiz, and exam items: • On any question that contains the key words \explain," \describe," \why," or other words that call for an explanation or interpretation, answer the question in complete sentence form. If your answer to the question involves quantities with units (like meters or seconds, for instance), make sure the units are included in your answer. • Show work on all questions that call for a calculation, regardless of whether or not the instructions in the problem ask you to show your work. Also make it clear what your final answer is; circling your answer is one way of doing this. Remember that your intermediate comments and calculations are just as important for grading purposes as your final answer. • On homework problems that you turn in for grading purposes, make sure that your work is neat and easy to follow. Work should not be crammed into small spaces, and items should not be scribbled out on the final draft that you turn in. Many problems require a lot of thought and trial and error, and some will take more space than you first expected. You may find that you need to do the problems first on scratch paper and then turn in a neater, rewritten final version. My policy is to warn you once about neatness and organization and to dock points if it continues. (continued on next page −!) 4 Sonoma State University In conclusion, I want to remind everyone that I am always happy to help you with any problems you might be having with the course, and I hope you will feel comfortable approaching me when you need this help. I look forward to working with all of you and hope that we will all have a great semester! Note. The following problems deal with review topics that should already be familiar to you from previous calculus and algebra courses. Being able to do problems similar to those below efficiently will be important to your success in Math 261. 1. Find the derivatives of each of the following with respect to x: You may assume that a is a constant: 2 2 1 1 (a) y = sin(xeax ) (b) y = sin(aexa ) (c) y = (d) y = x + 2a2 a + 2x2 2. Calculate the following integrals: Z e 1 Z 1 + e2x (a) 2 dx (d) x dx 1 x(1 + ln x) e Z Z ex (b) xex dx (e) dx 1 + e2x p Z sin( x) Z (c) p dx (f) (1 + ax2)2 dx x 3. Find the exact value of the area bounded by the curves y = ln x; y = 2; the x-axis, and the y-axis.