» Required Reading » Table of Contents » Chapter 14 14.3 Extremal Values on a Surface in Three Dimensions
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A surface in three dimensions is determined by one equation, which again we write as G = 0. Suppose, again that we wish to find extrema of F on this surface. This time F can have no non-vanishing component in the plane tangent to the surface an an extreme point, exactly as in the previous case. All this means that F and G must again point in the same direction. We can observe that this implies that the cross product FG must be 0, and this vector equation gives us two independent component equations that we can solve  along with G = 0 to find the extrema.

Also we can apply the Lagrange multiplier approach exactly as before. This time there are three components to all the vectors, so that the statement F = cG supplies us with three equations which along with G = 0 is enough to determine c and the coordinates of extrema.

Again you must identify maxima and minima and distinguish merely local from global at each extreme point.

When the surface is defined parametrically, you can compute gradients at each point and you need to use the gradient equations to determine the values of both parameters of the surface that determine the extreme points.