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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. |