14.4 Extrema on a Curve in Three Dimensions

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 on C

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.

This condition and G = 0 and H = 0 determine x, y and z at critical points.

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 and G = 0 and H = 0, to solve for c, d, x, y, and z.

Exercises:

14.6 Given a curve defined as the intersection of the surfaces defined by equations xyz = 1, and x² + 2y² +3z² = 7, find equations determining the critical points of 2x³ - y³ by the determinantal approach.

14.7 Write the equations for the critical points obtained using the Lagrange Multipliers approach for the same problem.

14.8 We seek the critical points for F on the curve x = 5 sin t, y = 3 cos 3t, z = sin 2t, for t = 0 to 2, with F = x² + y² + z². Write equations for them.