Home  18.013A  Chapter 14 
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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 nonzero 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 x^{2} + 2y^{2} +3z^{2} = 7, find equations determining the critical points of 2x^{3}  y^{3} 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 = x^{2} + y^{2} + z^{2}. Write equations for
them.
