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chain_rule

# chain_rule - Notes by David Groisser for MAS 4105 Copyright...

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Notes by David Groisser for MAS 4105, Copyright c 1998 Jacobians, Directional Derivatives, and the Chain Rule. Suppose f 1 , f 2 , ...f q are functions of p variables x 1 , ..., x p ; thus for 1 i q , f i ( x 1 , x 2 , ..., x p ) is some real number. For a given point x = ( x 1 , x 2 , ..., x p ) R p , we can assemble the numbers f 1 ( x ) , f 2 ( x ) , ..., f q ( x ) into a q -component column vector f(x) . (We will also write x as a column vector below.) Thus we obtain a map f : R p R q . (If one or more of the f i ’s is not defined at every point of R p , we actually get a function whose domain is just a subset of R p , not all of R p .) This is an important and sophisticated perspective on which much of advanced calculus is based. For what we do below, it is best to write points in R p and R q as column vectors, rather than row vectors. We call f : R p R q differentiable at a point a R p if all the partial derivatives ∂f i /∂x j (1 i q, 1 j p ) exist at x = a and are continuous there. (Technically this definition is not quite right, but it will suffice for us.) At each such a , we define a q × p matrix J f ( a ), called the Jacobian of f at a ; it is defined by ( J f ( a )) ij = ∂f i ∂x j ( a ) .

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chain_rule - Notes by David Groisser for MAS 4105 Copyright...

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