chap02 - differentiation - Differentiation 1 Chapter 2...

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Unformatted text preview: Differentiation 1 Chapter 2 Differentiation Now that we have a good understanding of the Euclidean space R n , we are ready to discuss the concept of differentiation in multivariable calculus. We are going to deal with functions defined on one Euclidean space with values in another Euclidean space. We shall use the shorthand notation R n f R m to describe such a situation. This means that f can be written in the form f ( x ) = ( f 1 ( x ) ,f 2 ( x ) ,...,f m ( x )) , where x = ( x 1 ,x 2 ,...,x n ) and each coordinate function f i is a real-valued function on R n . In these situations it may be that f is not defined on all of R n , but well continue with the above shorthand. There are two important special cases: n = 1 and m = 1, respectively. We shall quickly see that the case n = 1 is much, much simpler than all other cases. We shall also learn that the case m = 1 already contains almost all the interesting mathematics that we investigate the generalization to m > 1 will prove to be very easy indeed. A. Functions of one real variable (n = 1) In the situation R f R m we shall typically denote the real numbers in the domain of f by the letter t , and the points in R m in the usual manner by x = ( x 1 ,x 2 ,...,x m ). As we mentioned above, we can represent f in terms of its coordinate functions: f ( t ) = ( f 1 ( t ) ,f 2 ( t ) ,...,f m ( t )) . This formula displays the vector f ( t ) in terms of its coordinates, so that the function f can be regarded as comprised of m real-valued functions f 1 , f 2 ,...,f m . We often like to think of real-valued functions in terms of their graphs, but when m > 1 this viewpoint seems somewhat cumbersome. A more useful way to think of f in these higher dimensions is to imagine the points f ( t ) plotted in R m with regard to the independent variable t . In case f ( t ) depends continuously on t , the points f ( t ) then form some sort of continuous curve in R m : 2 Chapter 2 f t ( ) We have placed arrows on our picture to indicate the direction of increasing t . Thus the points f ( t ) form a sort of curve (whatever that may mean) in R m . We need to understand well the definition of limit as t t and/or continuity at t . As we are somehow interested in the size of f ( t )- f ( t ), we can merely use the definition of continuity in the case of real-valued functions, modified so that instead of absolute value we use the norm. Thus we have the DEFINITION . Let R f R m . Then f is continuous at t if for each > 0 there exists > such that | t- t | < k f ( t )- f ( t ) k < . PROBLEM 21. Let R f R m and let L R m . Write out the correct definition of lim t t ,t 6 = t f ( t ) = L....
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This note was uploaded on 09/28/2008 for the course MATH 1220 taught by Professor Hatcher during the Fall '08 term at Cornell University (Engineering School).

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chap02 - differentiation - Differentiation 1 Chapter 2...

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