14-5 - Math 200 Spring 2010 Handout 13 Section 14-5 The...

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Math 200, Spring 2010 Handout 13 Section 14-5 The Gradient and Directional Derivatives Definition of the Gradient Vector If f is a function of x and y , then the gradient of f is the vector function f (“del f ” or grad f ) defined by ( ) ( ) ( ) , , , , x y f f f x y f x y f x y x y = = + i j The gradient of a function ( ) , f x y at a point ( ) , P a b = is the vector ( ) ( ) ( ) , , , , P x y a b f f f a b f a b = ∇ = . In three variables, if ( ) , , P a b c = , ( ) ( ) ( ) , , , , , , , , P x y z f f a b c f a b c f a b c = . The gradient f “assigns” a vector P f to each point in the domain of f . 1. If ( ) 2 , f x y y x = , (a) find the gradient of f and (b) evaluate the gradient at point ( ) 1,2 P 2. Use the chain rule for gradients to find the gradient of ( ) , , xyz g x y z e = Theorem: Chain Rule for Paths If f is a differentiable function and ( ) ( ) ( ) ( ) ( ) , , t x t y t z t = c is a differentiable path, then ( ) ( ) ( ) ( ) t d f t f t dt = ∇ c c c i Explicitly, in the case of two variables, if ( ) ( ) ( ) ( ) , t x t y t = c is a differentiable path in 2 R , then ( ) ( ) ( ) ( ) ( ) ( ) , , t d f f f dx f dy f t f t x t y t dt x y x dt y dt = ∇ = = + c c c i i
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3. Suppose 2 ( , ) 3 f x y x xy = and ( ) ( )
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