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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 y-directions. The domain of ( ∇ f )( x, y ) is contained in the plane, and its values consists of vectors in the xy-plane. So its graph will consist of ‘arrows’ in the xy-plane. 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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