MATH 263D - Analytic Geometry and Calculus
Four Quarter Hours

I. PREREQUISITES
Successful completion of Math 263C or equivalent.

II. COURSE DESCRIPTION
The main topics to be studied in Math 263D are: functions of two or more variables; curves, surfaces, and vectors in three-space; partial derivatives and various Chain Rules; directional derivatives and gradients; methods for determining extreme values of functions on their domains; double integrals in Cartesian and polar coordinates; triple integrals in Cartesian, cylindrical, and spherical coordinates.

The material presented is well-suited for students in mathematics, engineering, and the sciences.

III. TEXTBOOK AND CALCULATOR
0-13-518911-X Varberg, Dale, and Purcell, Edwin J., Calculus, 7th ed., Upper Saddle River, New Jersey: Prentice Hall, Inc., 1997.

Use of a calculator is optional in this course, but you may find it helpful. A calculator with basic arithmetic and square root functions is sufficient. Programmed calculators MAY NOT be used.

...available from EdMap's distance-learning online bookstore.

IV. COURSE CONTENT
Math 263D covers sections 14.5, 14.6, 14.7 of Chapter 14, and Chapters 15, 16. The following sections are omitted: 16.5, 16.6, and in Section 16.8, the portions of Example 1 and Example 3 having to do with finding the mass or center of mass of a solid. Also, in Section 16.8, omit the topic "Change of Variables in Multiple Integrals", pages 816 and 817. In Section 14.5, omit the material beginning with "Curvature" at the bottom of page 691 and extending through page 694. You will probably find the reading material in Section 15.4, "Differentiability" rather difficult, including the definition on page 732 of what it means to say a function f is differentiable at a point p = (x, y), and the definition of the gradient vector  at a point p. In any event, your reading in Section 15.4 should focus on knowing the conclusions of Theorem A and B, page 733.

The conclusion in Theorem A, which can be stated as:

for a function f of two variables x and y, and as

for a function f of three variables x, y, z, gives a practical rule for computing gradients. Theorem B gives sufficient conditions for a function f to be differentiable at a point p. As rated in the textbook, Theorems A and B together provide just what is needed to handle most problems involving differentiability and gradients.

In Section 15.6, it should be helpful for you to study the various examples to see how to apply the Chain Rules stated in Theorem A, page 743, and Theorem B, page 745. Also, note carefully Example 6, page 746 and Example 7, page 747, illustrating implicit differentiation for functions of two and three variables, respectively.

It will not be necessary for the purposes of this course to work problems from section 15.3, "Limits and Continuity." However, the section is important in mathematical theory, and should be read.

V. EXAMINATION
The examination for Math 263D in this format is a comprehensive supervised examination consisting of problems covering the material you are responsible for in Chapters 14, 15, and 16. The problems in the exam are similar to the basic problems in the textbook, and very similar to those in the Sample Examination.

You will have three hours to complete the supervised examination. A hand-held calculator will not really be needed, but may be used on the examination if you supply your own. Programmable calculators MAY NOT be used.

Note: You will not be required in the examination to prove theorems. However, it is important for you to know what the theorems say, and be able to apply them.

VI. PREPARING FOR THE EXAMINATION
Probably the most effective way for you to prepare for the examination is to concentrate on solving problems (especially the non-proof type problems) from the problem sets of each assigned section. However, for the necessary background to solve problems, you will need to understand the concepts and important results in each section before you attempt problem solving. Thus, for each assigned section, it is recommended that you first read carefully the material in the section and work through the details of each sample.

Theorems and important definitions are set off in rectangular boxes and should be learned. It is also recommended that you do the "Concepts Review" problems in each section to test your understanding of the material presented in the section. Answers to the Concept Review problems are given at the end of the section.

After you have studied the material presented in a section, you should solidify your understanding by working typical problems from the problem set for the section. Some of the higher numbered problems in the problem sets are quite hard, or of the "proof type," and you will not be held responsible for working these kind of problems on the supervised examination.

VII. SAMPLE EXAMINATION
A self-check Sample Examination is included with this syllabus. There is also an answer key which gives the answers to all problems in the sample examination. The solutions to these problems are given in considerable detail, so that you can follow completely the steps and methods required to solve each problem. The sample exam is provided to help you make a final determination as to whether or not you are prepared to take the supervised examination.

It is recommended that you do not take the sample examination until you feel you have adequately mastered the course material. When you do feel prepared to take the sample exam, take it without using books or notes. Finally, evaluate your answers using the answer key. Sample Examination

If you do not do well on the sample examination, then review more extensively those troublesome areas before applying for the examination. As mentioned earlier, the problems on the supervised examination will be quite similar to those on the sample exam. Hence, it should also be helpful to you in preparing for the supervised examination to look closely at the sample exam answer key solutions to those problems you answered incorrectly on the sample exam.

IX. GRADING SCALE
Your course grade will be determined by your percent score on the examination, using the following scale:

88-100% = A to A
75-87% = B to B+
60-74% = C to C+
50-59% = D to D+
Below 50% = F