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Unformatted text preview: Chapter 2 Differentiation in higher dimensions 2.1 The Total Derivative Recall that if f : R R is a 1-variable function, and a R , we say that f is differentiable at x = a if and only if the ratio f ( a + h )- f ( a ) h tends to a finite limit, denoted f ( a ), as h tends to 0. There are two possible ways to generalize this for vector fields f : D R m , D R n , for points a in the interior D of D . (The interior of a set X is defined to be the subset X obtained by removing all the boundary points. Since every point of X is an interior point, it is open.) The reader seeing this material for the first time will be well advised to stick to vector fields f with domain all of R n in the beginning. Even in the one dimensional case, if a function is defined on a closed interval [ a,b ], say, then one can properly speak of differentiability only at points in the open interval ( a,b ). The first thing one might do is to fix a vector v in R n and saythat f is differentiable along v iff the following limit makes sense: lim h 1 h ( f ( a + hv )- f ( a )) . When it does, we write f ( a ; v ) for the limit. Note that this definition makes sense because a is an interior point. Indeed, under this hypothesis, D contains a basic open set U containing a , and so a + hv will, for small enough h , fall into U , allowing us to speak of f ( a + hv ). This 1 derivative behaves exactly like the one variable derivative and has analogous properties. For example, we have the following Theorem 1 (Mean Value Theorem for scalar fields) Suppose f is a scalar field. Assume f ( a + tv ; v ) exists for all t 1 . Then there is a t o with t o 1 for which f ( a + v )- f ( a ) = f ( a + t v ; v ) . Proof . Put ( t ) = f ( a + tv ). By hypothesis, is differentiable at every t in [0 , 1], and ( t ) = f ( a + tv ; v ). By the one variable mean value theorem, there exists a t such that ( t ) is (1)- (0), which equals f ( a + v )- f ( a ). Done. When v is a unit vector , f ( a ; v ) is called the directional derivative of f at a in the direction of v . The disadvantage of this construction is that it forces us to study the change of f in one direction at a time. So we revisit the one-dimensional definition and note that the condition for differentiability there is equivalent to requiring that there exists a constant c (= f ( a )), such that lim h f ( a + h )- f ( a )- ch h = 0. If we put L ( h ) = f ( a ) h , then L : R R is clearly a linear map. We generalize this idea in higher dimensions as follows: Definition. Let f : D R m ( D R n ) be a vector field and a an interior point of D . Then f is differentiable at x = a if and only if there exists a linear map L : R n R m such that ( * ) lim u || f ( a + u )- f ( a )- L ( u ) || || u || = 0 ....
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This note was uploaded on 04/18/2008 for the course MA 1C taught by Professor Ramakrishnan during the Spring '08 term at Caltech.
- Spring '08