partial_derivatives.nb

Partial Derivatives

We define F to be a function of two variables whose graph lies on the unit sphere:

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Here are its partial derivatives

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We define functions representing its partial derivatives:

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Fx[x_, y_] := -x/(1 - x^2 - y^2)^(1/2) Fy[x_, y_] := -y/(1 - x^2 - y^2)^(1/2)

Here is the graph:

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[Graphics:HTMLFiles/index_9.gif]

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(This is code to draw the tangent vectors obtained from the partial derivatives)

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$lx[x_, y_] := Graphics3D[{RGBColor[1, 0, 0], Thickness[.01], Line[{{x, y, F[x, y]}, {x, y, F[x ... GBColor[0, 1, 0], Thickness[.01], Line[{{x, y, F[x, y]}, {x, y, F[x, y]} + {0, 1, Fy[x, y]}/2}]}]$

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$TangentVector[x_, y_] := {p1, lx[x, y], ly[x, y]}$

This shows the tangent vectors <1,0,Fx[x,y]> in red and <0,1,Fy[x,y]> based at the point (x,y)=(0,0)

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[Graphics:HTMLFiles/index_14.gif]

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at (x, y)=(0.4, 0)

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[Graphics:HTMLFiles/index_17.gif]

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At (x, y)=(0.5, 0.5)

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[Graphics:HTMLFiles/index_20.gif]

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Created by Mathematica (September 27, 2004)