   Chapter 16.6, Problem 2E

Chapter
Section
Textbook Problem

Determine whether the points P and Q lie on the given surface.2. r(u, v) = ⟨1 + u − v, u + v2, u2 – v2⟩ P(1, 2, 1), Q(2, 3, 3)

To determine

Whether the point P(1,2,1) lies on the surface r(u,v)=1+uv,u+v2,u2v2 or not and whether the point Q(2,3,3) lies on the surface r(u,v)=1+uv,u+v2,u2v2 or not.

Explanation

Given data:

The surface vector is r(u,v)=1+uv,u+v2,u2v2 and the point is P(1,2,1) .

The surface vector is r(u,v)=1+uv,u+v2,u2v2 and the point is Q(2,3,3) .

The point P(1,2,1) lies on the surface r(u,v)=1+uv,u+v2,u2v2 if and only if there are values for u and v such that the corresponding values of u and v must satisfy the vector r(u,v)=1+uv,u+v2,u2v2 .

Equate x-, y-, and z-coordinates of the point to the corresponding points of the vector r(u,v) and obtain the values of u and v as follows.

1+uv=1 (1)

u+v2=2 (2)

u2v2=1 (3)

Simplify the equation (1) as follows.

uv=11uv=0u=v

Substitute v for u in equation (3),

v2v2=101

The values of u and v are not satisfied for the vector r(u,v) .

Thus, the point P(1,2,1) does not lie on the surface

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