chap1_update1 - 1 Dierentiability and the calculus of...

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1 Di ff erentiability and the calculus of variations We begin with some preliminaries on di ff erentiation 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 infinite dimensional case and the calculus of variations. 1.1 Di ff erentiation 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 , . . . , x n ) . . . g m ( x 1 , . . . , x n ) , where g i ( x 1 , . . . , x n ) are scalar-valued functions. Such a function is called di ff erentiable if it is locally approximable by a linear function, i.e. by a matrix. More precisely, g is di ff erentiable 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 definition of di ff erentiability 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 ) := g 1 x 1 x ) . . . g 1 x n x ) . . . . . . g m x 1 x ) . . . g m x 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 Di ff erentiability 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 , defined 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 ff eren- tiable functional 1 . The directional derivative of g at ¯ x R n in the direction v R n is defined by d d α α =0 g x + α v ) . The function g x + α v ) 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 + α v ) : 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 d α α =0 g x + α v ) = ( δ g x )) ( δ
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