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lect12_6

lect12_6 - The Gradient and Directional Derivative(12.6 1...

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The Gradient and Directional Derivative - (12.6) 1. Directional Derivatives Definition: The directional derivative of f ! x , y " at the point ! a , b " and in the direction of the unit vector u ! "# u 1 , u 2 \$ , denoted as D u ! f ! a , b " , is defined by D u ! f ! a , b " " lim h % 0 f ! a & hu 1 , b & hu 2 " ! f ! a , b " h provided partial derivatives exist. Note that ! If u ! "# 1, 0 \$ , D u ! f ! a , b " " lim h % 0 f ! a & h , b " ! f ! a , b " h " f x and if u ! "# 0,1 \$ , D u ! f ! a , b " " lim h % 0 f ! a , b & h " ! f ! a , b " h " f y ! If D u ! f ! a , b " \$ 0, then f ! x , y " is increasing at ! a , b " and in the direction u ! . If D u ! f ! a , b " # 0, then f ! x , y " is decreasing at ! a , b " and in the direction u ! . Property: Suppose that f is differentiable at ! a , b " and u ! "# u 1 , u 2 \$ is any unit vector. Then D u ! f ! a , b " " f x ! a , b " u 1 & f y ! a , b " u 2 . Let g ! h " " f ! a & hu 1 , b & hu 2 " . Then g ! 0 " " f ! a , b " . So, D u ! f ! a , b " " lim h % 0 f ! a & hu 1 , b & hu 2 " ! f ! a , b " h " lim h % 0 g ! h " ! g ! 0 " h " g ! 0 " Let x ! h " " a & hu 1 , y ! h " " b & hu 2 . g ! h " " f x dx dh & f y dy dh " f x ! a & hu 1 , b & hu 2 " u 1 & f y ! a & hu 1 , b & hu 2 " u 2 g ! 0 " " f x ! a , b " u 1 & f y ! a , b " u 2 . In a similar way, we can derive the directional derivative of f ! x , y , z " at ! a , b , c " and in the unit direction u ! "# u 1 , u 2 , u 3 \$ D u ! f ! a , b , c " " f x ! a , b , c " u 1 & f y ! a , b , c " u 2 & f z !

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