   Chapter 13.1, Problem 45E

Chapter
Section
Textbook Problem

# Find a vector function that represents the curve of intersection of the two surfaces.45. The hyperboloid z = x2 − y2 and the cylinder x2 + y2 = 1

To determine

To find: the vector function of a curve that intersects two surfaces.

Explanation

Given data:

The equation of the hyperboloid is z=x2y2 and the cylinder is x2+y2=1 .

Definition:

Consider parametric equations as x=f(t) , y=g(t) , and z=h(t) .

Write the expression for vector function in terms of parametric equations.

r(t)=f(t)i+g(t)j+h(t)k (1)

Here,

f(t) , g(t) , and h(t) are components functions of r(t) , and

Modify equation (1) as follows.

r(t)=xi+yj+zk (2)

The surface equation x2+y2=1 is a circle with the radius as 1. Since, z-plane is at z=0 . Consider x=cost and y=sint with the interval 0t2π .

Write the expression for surface of cylinder.

x2+y2=1

Substitute cost for x and sint for y,

(cost)2+(sint)2=1cos2t+sin2t=1 {cos2x+sin2x=1}1=1

The RHS is equal to the LHS

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