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Unformatted text preview: LIST OF TOPICS COVERED IN CHAPTER 14 JORIS VANKERSCHAVER Abstract. This is a summary of the definitions and results of chapter 14 that we discussed during class. Whenever a proof is provided, I’ve taken care to present only the essential idea, without all of the supplementary details; you can consult the textbook for additional discussions and more examples. These notes only provide you an outline of the material that we covered in class and should be seen as complementary to the textbook. In particular, reading these notes is not a substitute for attending class, studying the text book and solving problem sets! Please signal any mistakes, omissions, etc. to [email protected] . 1. Note on Midterm #2 Midterm #2 will cover sections 13.3 and 13.5 , which are not described in this document, together with sections 14.1-7 . The emphasis will be on chapter 14, but keep in mind that a lot of the calculations in that chapter rely on techniques that were covered earlier (e.g. you can’t really do any computations involving the tangent plane if you have no idea of what a plane is). 2. Note on Graphing Functions For complicated functions of two variables, it is often helpful to obtain some understanding of how the function looks like by means of a computer plot. There are currently a number of free and easy-to-use plotting systems, the easiest being Wolfram Alpha ( http://www. wolframalpha.com ). This system has the advantage that it accepts plain English as its input, and it does a reasonable job of producing good-looking figures. The figures in this document were all made with Wolfram Alpha, and the commands needed to reproduce them are described in the pictures. Keep in mind, however, that you will not have access to a computer during the exams! 3. Functions of Two or More Variables • A function of n variables is an expression f ( x 1 ,...,x n ) which gives you a real number when you plug in values for the variables x 1 ,...,x n . The domain of a function of n variables is the set of values D ⊂ R n for which the function is defined. The range is the set of (real) values that the function is able to attain. (1) Let f ( x,y ) = 3 x 2 y . This is a function of two variables, the domain is the whole of R 2 , and the range is R . The latter can be seen (for instance) by fixing x = 1, so that f (1 ,y ) = 3 y . As y varies over the real numbers, so does f (1 ,y ). 1 2 JORIS VANKERSCHAVER (2) Consider the following function of three variables: f ( x,y,z ) = sin xy z . Show that the domain is D = ( x,y,z ) ∈ R 3 : z 6 = 0 , and that the range is the interval [- 1 , 1]. (3) Same question for f ( x,y ) = log x 2 y . Use the figure below to corroborate your answer....
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This note was uploaded on 09/13/2011 for the course MATH 20C taught by Professor Helton during the Spring '08 term at UCSD.
- Spring '08
- Multivariable Calculus