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Gradients

We must load the vector field plotting package

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Here is a simple function:

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Here are it's partial derivatives:

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Here we define a function that gives us the gradient vector

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$gradf[x_, y_] := {fx[x, y], fy[x, y]}$

An example

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Here we plot the vector field determined by the function

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[Graphics:HTMLFiles/index_13.gif]

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[Graphics:HTMLFiles/index_16.gif]

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[Graphics:HTMLFiles/index_19.gif]

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Here is a graph of our function

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[Graphics:HTMLFiles/index_22.gif]

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This is the norm of the gradient vector for this function. It is a cone.

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[Graphics:HTMLFiles/index_25.gif]

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Another view

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[Graphics:HTMLFiles/index_28.gif]

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Another example

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[Graphics:HTMLFiles/index_34.gif]

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The partial derivatives

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A function defining the gradient

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$gradf[x_, y_] := {Cos[x] Sin[y], Cos[y] Sin[x]}$

Here is the gradient vector field

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[Graphics:HTMLFiles/index_41.gif]

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Here is the gradient vector field shown with more vectors plotted

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[Graphics:HTMLFiles/index_44.gif]

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Here is the countour plot of our surface

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[Graphics:HTMLFiles/index_47.gif]

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The gradient vector field and contour plot are shown together.

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[Graphics:HTMLFiles/index_50.gif]

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The norm of the gradient is graphed below

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[Graphics:HTMLFiles/index_53.gif]

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The graph of our function

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[Graphics:HTMLFiles/index_56.gif]

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Here they are shown together

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[Graphics:HTMLFiles/index_59.gif]

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Created by Mathematica (October 4, 2004)