Home  18.013A  Chapter 14 


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 $\stackrel{\u27f6}{\nabla}F\xb7\stackrel{\u27f6}{t}$ 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 $\stackrel{\u27f6}{\nabla}F$ 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 $H=0,\stackrel{\u27f6}{t}$ is in the direction of $\stackrel{\u27f6}{\nabla}G\times \stackrel{\u27f6}{\nabla}H$ , and the statement that $\stackrel{\u27f6}{\nabla}F$ has no component in that direction is the statement that $\stackrel{\u27f6}{\nabla}F$ lies in the plane of $\stackrel{\u27f6}{\nabla}G$ and $\stackrel{\u27f6}{\nabla}H$ $$ 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 $\stackrel{\u27f6}{\nabla}F=c\stackrel{\u27f6}{\nabla}G+d\stackrel{\u27f6}{\nabla}H$ . 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 ${x}^{2}+2{y}^{2}+3{z}^{2}=7$ , find equations determining the critical points of $2{x}^{3}{y}^{3}$ 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
$x=5\mathrm{sin}t,y=3\mathrm{cos}3t,z=\mathrm{sin}2t$
, for
$t=0$
to
$2\pi $
, with
$F={x}^{2}+{y}^{2}+{z}^{2}$
. Write equations for them.
