Find a vector function that represents the curve of the intersection of the two surfaces. The cylinder x² + y² = 4 and the surface z = xy.

Solution:

Given, the equation of the cylinder is x² + y² = 4

Given, the equation of the surface is z = xy

We have to find the vector function that represents the curve of the intersection of the two surfaces.

A vector function is represented as

r(t) = x_ti +y_tj + z_tk --- (1)

The equation of cylinder can be written as x² + y² = (2)² × 1

From trigonometric identity,

cos²t + sin²t = 1

Substituting the trigonometric identity,

x² + y² = (2)²[cos²t + sin²t]

x² + y² = 2² cos²t + 2² sin²t

x² + y² = [2 cost]² + [2 sint]²

By comparison,

xv = [2 cost]²

x = 2 cost

y² = [2 sint]²

y = 2 sint

Also, z = xy

z = (2 cost)(2 sint)

z = 4 cost sint

Substituting the values of x, y and z in (1),

r(t) = (2 cost)i + (2 sint)j + (4 cost sint)i

Therefore, the vector function that represents the intersection of the two surfaces is

r(t) = (2 cost)i + (2 sint)j + (4 cost sint)i.

---

Find a vector function that represents the curve of the intersection of the two surfaces. The cylinder x² + y² = 4 and the surface z = xy.

Summary:

A vector function that represents the curve of intersection of the two surfaces. The cylinder x² + y² = 4 and the surface z = xy is r(t) = (2 cost)i + (2 sint)j + (4 cost sint)i.