Unformatted text preview: mappings allows us to formulate a single uniFed version which includes the others as special cases. In the statements below, we assume the dimensions m , n and p are each 1, 2 or 3. Theorem 4.2.5 (General Chain Rule) . If F : R n → R m is diFerentiable at −→ y ∈ R n and G : R p → R n is diFerentiable at −→ x ∈ R p where −→ y = G ( −→ x ) , then the composition F ◦ G : R p → R m is diFerentiable at −→ x , and its derivative is the composition of the derivatives of G (at −→ x ) and F (at −→ y = G ( −→ x ) ): D ( F ◦ G ) −→ x = ( DF −→ y ) ◦ ( DG −→ x ); 5 Again, the analogue when the domain or the target or both are the plane instead of space is straightforward. 6 An older, but sometimes useful notation, based on viewing F as an mtuple of functions, is ∂ ( f 1 ,f 2 ,f 3 ) ∂ ( x,y,z ) ....
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 Spring '08
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 Calculus, Chain Rule, Derivative, Transformations, representative, th partial derivatives

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