This preview shows pages 1–3. Sign up to view the full content.
This preview has intentionally blurred sections. Sign up to view the full version.
View Full Document
Unformatted text preview: Chapter 2 Differentiation in higher dimensions 2.1 The Total Derivative Recall that if f : R R is a 1variable 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 onedimensional 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 ....
View
Full
Document
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
 Ramakrishnan
 Derivative

Click to edit the document details