20e-lecture7

20e-lecture7 - Lecture 7 Monday April 13th [email protected]..

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Unformatted text preview: Lecture 7 : Monday April 13th [email protected] Key concepts : C k ( R n ) , C ∞ ( R n ) , polynomials, mixed partial derivatives Know how to find higher order partial derivatives 7.1 Higher partial derivatives The partial derivatives we have studied so far have meaning in terms of the slope of surfaces in various directions. If these partial derivatives as functions themselves are differentiable, we can obtain second order partial derivatives. These have very important meaning when it comes to continuous optimization, as we shall see later. For now we concentrate on the mechanics of higher order partial derivatives. A function f : R n → R is in class C 1 if ∇ f : R n → R n is a continuous function. In other words, all the partial derivatives of f exist and are continuous. If all the partial derivatives of f themselves have continuous derivatives, then we say f is in class C 2 . For example, if f : R 2 → R then ∇ f = ( f x ,f y ) if it exists, and if f x and f y have derivatives with respect to x and y , then these derivatives are written f xx ,f xy ,f yx ,f yy . Alternatively, we write these derivatives as f xx = ∂ 2 f ∂x 2 f yx = ∂ 2 f ∂x∂y f xy = ∂ 2 f ∂y∂x f yy = ∂ 2 f ∂y 2 . It is very important to note that in general f xy 6 = f yx . For practical purposes, ∂ 2 f ∂x∂y = ∂ ∂x ∂f ∂y ¶ so derivatives are worked out from right to left in this notation. In the notation f...
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This note was uploaded on 02/12/2010 for the course MATH 20E taught by Professor Enright during the Spring '07 term at UCSD.

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20e-lecture7 - Lecture 7 Monday April 13th [email protected]..

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