How can I find a vector function that represents the curve of intersection between the cylinder defined by x^2 + y^2 = 4 and the surface z = xy?

To find the vector function that represents the curve of intersection between the cylinder x² + y² = 4 and the surface z = xy, we first need to express the points on the cylinder in a suitable parametric form.

The equation of the cylinder describes a circle of radius 2 in the xy-plane. We can parametrize this circle using a parameter t as follows:

x(t) = 2 	imes cos(t)

y(t) = 2 	imes sin(t)

Next, we can use these expressions to find z by substituting x(t) and y(t) into the equation for the surface:

z(t) = x(t) 	imes y(t) = (2 	imes cos(t)) 	imes (2 	imes sin(t))
       = 4 	imes cos(t) 	imes sin(t)

Using the trigonometric identity for sine, we can rewrite z(t):

z(t) = 2 	imes sin(2t)

Now we have the parametric equations for the curve of intersection:

• x(t) = 2 imes cos(t)
• y(t) = 2 imes sin(t)
• z(t) = 2 imes sin(2t)

Combining these, we can write the vector function r(t) that represents the curve:

r(t) = { 2 	imes cos(t), 2 	imes sin(t), 2 	imes sin(2t) }

Where the parameter t ranges from 0 to 2 ext{π}.

This vector function r(t) will adequately represent the curve of intersection between the specified cylinder and the surface across the parameter interval. The use of parametric equations helps in visualizing and further analyzing the intersection in 3D space.