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Unformatted text preview: Differentiable Functions Let S â R n be open and let f : R n â R . We recall that, for x o = ( x o 1 , x o 2 , Â·Â·Â· , x o n ) â S the partial derivative of f at the point x o with respect to the component x j is defined as âf ( x o ) âx j := lim h â f ( x o 1 , x o 2 , Â·Â·Â· , , x o j- 1 , x o j + h, x o j +1 , Â·Â·Â· , x o n ) h , provided this limit exists. If this limit exists for the points x â S , then we can differentiate the resulting function x 7â âf ( x ) âx j with respect to any of the components of x to obtain â 2 f ( x ) âx k âx j , the second partial derivative of f . In particular, if f i 6 = j we refer to this second partial derivative as the mixed partial derivative . An important property of this mixed partial derivative is that â 2 âx k âx j f ( x ) = â 2 âx j âx k f ( x ) , provided these second derivatives exist and are continuous. Higher order derivitives are defined in a similar manner. A real-valued function f : S â R will be said to be of class C ( k ) on the open set S provided it is continuous and posesses continuous partial derivatives of all orders up to and including k . It will be said to be of class C ( â ) on S if it is of class C ( k ) for all integers k . It will be said to be of class C ( k ) on an arbitrary set S provided it is of class C ( k ) on a neighborhood of that set. We will also use the notation C and D 00 for the classes C (1) and C (2) respectively. Alternate notations for the partial derivatives will also be used, for example, f x i = âf âx j , f x k ,x j = â 2 âx k âx j f , etc. The notion of differentiability of functions of several variables is related to the existence of partial derivatives but is not coincident with the existence of the partials. Indeed, we have the following definition: A function f : S â R m where S â R n is an open set, is said to be differentiable at a point x o â S provided there is a linear transformation L : R n â R m such that, for all h â R n lim k h kâ f ( x o + h )- f ( x o )- L h k h k = 0 . In the case that such a linear transformation L exists, it is called the derivative (sometimes the FrÂ´ echet derivative or the differential) of the function f at the point x o . There are 1 various notations for the differential. Since the linear transformation depends on the point x o we may denote it as L = L ( f ; x o ) when we need to be specific. Other notations will be f ( x o , h ) = L ( f ; x o )( h ). Now, in the case that f : S â R , the linear transformation is a linear map from R n to R and is therefore called a a linear functional . This linear functional can be realized by the application of a dot product. Given any fixed vector z â R n it is clear that the map of R n â R given by ` z ( h ) := z Â· h = z >...
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- Fall '10
- Derivative, XO