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 $\stackrel{⟶}{\nabla }F·\stackrel{⟶}{t}$ 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 $\stackrel{⟶}{\nabla }F$ 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 $H=0,\stackrel{⟶}{t}$ is in the direction of $\stackrel{⟶}{\nabla }G\times \stackrel{⟶}{\nabla }H$ , and the statement that $\stackrel{⟶}{\nabla }F$ has no component in that direction is the statement that $\stackrel{⟶}{\nabla }F$ lies in the plane of $\stackrel{⟶}{\nabla }G$ and $\stackrel{⟶}{\nabla }H$ $$ 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 $\stackrel{⟶}{\nabla }F=c\stackrel{⟶}{\nabla }G+d\stackrel{⟶}{\nabla }H$ . 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 $2\pi$, with $F=x^2+y^2+z^2$. Write equations for them.