]> 14.3 Extremal Values on a Surface in Three Dimensions  14.3
Extremal Values on a Surface in Three Dimensions  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 extreme point,
exactly as in the previous case.  This means that F and G must
again point in the same direction.  We can observe that this
implies that the cross product F × G 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 Multipliers approach exactly as before. This
time there are three components to all the vectors, so that the
statement F = c G supplies us with three equations which, along
with G = 0 , is enough to determine c and the coordinates of the
extrema.  Again you must identify maxima and minima and
distinguish merely local