» Required Reading » Table of Contents » Chapter 14 14.4 On a Curve in Three Dimensions
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A curve C in three dimensions can be defined by two equations (that is as the intersection of two surfaces) or by use of a single parameter as in two dimensions.
If q is an  extreme values of F on C we cannot have Ft non-zero at argument q, by our general principle; otherwise F will be larger on one side of q and smaller on the other than its value at q.

The implications of this condition are different here however. We can no longer say that F points in some particular direction at an extremal point. Rather it must be normal to some particular direction, that of the tangent vector to C at such points.

When C is described by two equations, G = 0 and H = 0, t is in the direction of G H, and the statement that F has no component in that direction is the statement that F lies in the plane of G and H and so the volume of their parallelepiped is 0 and the determinant whose columns are all these grads must be 0.

Another way to state the same condition is to use two Lagrange Multipliers, say c and d: and write F = c G + d H.  We can solve the three equations obtained by writing all three components of this vector equation and use them to solve for c, d and one more variable, which is in general all that you need  to locate the extremal points on the curve.