12.6 Notes - f in the direction of the unit vector u is •...

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12.6—Directional Derivatives and Gradients Definition of Directional Derivative Let f be a function of two variables x and y and let u = θ cos i + sin j be a unit vector. Then the directional derivative of f in the direction of u , denoted by D ) , ( y x f u is t y x f t y t x f y x f D t u ) , ( ) sin , cos ( lim ) , ( 0 - + + = provided this limit exists. Theorem 12.9 Directional Derivatives If f is a differentiable function of x and y, then the directional derivative of f is in the direction of the unit vector u = cos i + sin j is sin ) , ( cos ) , ( ) , ( y x f y x f y x f D y x u + = #4 #14 Definition of Gradient of a Function of Two Variables Let ) , ( y x f z = be a function of x and y such that x f and y f exist. Then the gradient of f, denoted by ) , ( ) , ( y x f y x f x = i + ) , ( y x f y j Read f as “del f “. Another notation for the gradient is grad ) , ( y x f . Keep in mind that for each (x,y), the gradient ) , ( y x f is a vector in the plane, not in space. #26
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Theorem 12.10 Alternative Form of the Directional Derivative If f is a differentiable function of x and y, then the directional derivative of
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Unformatted text preview: f in the direction of the unit vector u is • ∇ = ) , ( ) , ( y x f y x f D u u #28 Properties of the Gradient Let f be differentiable at the point (x,y) 1. If ) , ( y x f ∇ = then D ) , ( y x f u =0 for all u 2. The direction of maximum increase of f is given by ) , ( y x f ∇ . The maximum value of D ) , ( y x f u =0 is ) , ( y x f ∇ . 3. The direction of minimum increase of f is given by - ) , ( y x f ∇ . Then minimum value of D ) , ( y x f u is - ) , ( y x f ∇ . #32 Theorem 12.12 Gradient is Normal to Level Curves If f is differentiable at ( 29 , y x and f ∇ ( 29 , y x ≠ 0 then f ∇ ( 29 , y x is normal to the level curve through ( 29 , y x . #56 Note: The definitions and properties of directional derivatives and gradients can be extended to functions of three or more variables as seen on page 892....
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12.6 Notes - f in the direction of the unit vector u is •...

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