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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 ∇ ⟶ F · 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 ∇ ⟶ 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 x y z = 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 sin ⁡ t , y = 3 cos ⁡ 3 t , z = sin ⁡ 2 t , for t = 0 to 2 π , with F = x 2 + y 2 + z 2 . Write equations for them.

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