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chap1_update1 - 1 Dierentiability and the calculus of...

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1 Diferentiability and the calculus oF variations We begin with some preliminaries on diferentiation oF Functions in R n . Read- ers Familiar with this material should still skim over this section as the con- cepts are presented in a manner that immediately generalizes to the in±nite dimensional case and the calculus oF variations. 1.1 Diferentiation in R n Let g be a vector valued Function oF several variables. We write g : R n -→ R m to denote that g is a Function oF n variables x =( x 1 ,...,x n ), and produces m × 1 vectors. Thus g ( x ) is short hand For g ( x )= g 1 ( x 1 n ) . . . g m ( x 1 n ) , where g i ( x 1 n ) are scalar-valued Functions. Such a Function is called diferentiable iF it is locally approximable by a linear Function, i.e. by a matrix. More precisely, g is diferentiable at a point ¯ x R n iF there exists a matrix D such that lim v R n , ± v ±→ 0 ± ( g x + v ) - g x )) - Dv ± ± v ± =0 . (1.1) It can be shown that this de±nition oF diferentiability is equivalent to the requirement that all partial derivatives ∂g i ∂x j x ) exist and are continuous at ¯ x . In this case, the matrix approximation oF g at ¯ x (i.e. the matrix D above) is given by the Jacobian (denoted by δ g x )) oF g at ¯ x δ g x ) := 1 1 x ) ... 1 n x ) . . . . . . m 1 x ) m n x ) . (1.2) Thus For any Function g ( . ): R n R m , its derivative is a Function that associates with every point in R n a linear mapping From R n R m namely
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2 1 Diferentiability and the calculus oF variations the Jacobian, i.e. δ g ( . ): R n -→ R m × n , where R m × n is the space of m × n matrices. In other words, the derivative of a vector-valued function is a matrix -valued function. When two multivariable functions are composed, the derivative of the composition can be computed from a version of the chain rule which we now state. If h : R n R m and g : R m R q , then their composition f := ( g h R n R q , deFned by x R n , f ( x ) := g ( h ( x )) has the derivative δ f x ) = ( δ g ( h x ))) ( δ h x )) , (1.3) where the product on the right is matrix multiplication. The chain rule can be understood as stating that at any point, the derivative of a composition is the composition of the derivatives (the composition of the derivatives is given by matrix multiplication since the derivative at a point is a linear mapping). As an illustration of the chain rule, we compute a formula for the direc- tional derivatives of scalar valued functions. Let g : R n R be a di±eren- tiable functional 1 . The directional derivative of g at ¯ x R n in the direction v R n is deFned by d ± ± ± ± α =0 g x + αv ) . The function g x + αv ) can be seen as the composition of two functions, namely g and another function h : R R n . Since α is a scalar and v R n , the function h ( α ) := (¯ x + αv R R n can be visualized as a vector in R n starting at ¯ x and with length α in the direction v . The derivative of this function is δ h = v , and the chain rule implies d ± ± ± ± α =0 g x + αv )=( δ g x )) ( δ h (0)) = ² ∂g ∂x 1 x ) ...
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chap1_update1 - 1 Dierentiability and the calculus of...

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