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Unformatted text preview: Gradients, Directional Derivatives John E. Gilbert, Heather Van Ligten, and Benni Goetz What’s going to replace the derivative f ( x ) of a function y = f ( x ) of one variable when z = f ( x, y ) depends on two or more variables? Since slope now depends on direction, not just sign, we can expect vectors will be needed! Definition: the Gradient of z = f ( x, y ) is the vector function ( ∇ f )( x, y ) = ∂f ∂x i + ∂f ∂y j whose components are the partial derivatives of f in the x and ydirections. The domain of ( ∇ f )( x, y ) is contained in the plane, and its values consists of vectors in the xyplane. So its graph will consist of ‘arrows’ in the xyplane. In later courses you’ll call such functions vector fields . Example: for the hyperbolic paraboloid z = f ( x, y ) = x 2 y 2 , its gradient is ( ∇ f )( x, y ) = 2 x i 2 y j . The graph of ∇ f is shown to the right. Notice how the length of the vectors increases as ( x, y ) moves away from the origin. How is the length and direction of these vectors connected with the graph of the hyperbolic paraboloid z = x 2 y 2 ?...
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This note was uploaded on 02/13/2012 for the course M 408 D taught by Professor Textbookanswers during the Spring '07 term at University of Texas.
 Spring '07
 TextbookAnswers
 Derivative, Slope

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