How do I find a vector function representing the curve of intersection of the cylinder x^2 + y^2 = 4 and the surface z = xy?

To find a vector function that represents the curve of intersection of the two given surfaces, we need to analyze both equations:

• Cylinder: The equation of the cylinder is given by x² + y² = 4, which represents a circular cylinder with a radius of 2 centered along the z-axis.
• Surface: The surface is defined by z = xy.

To express the curve of intersection, we can use a parameterization for the cylinder. A common choice is to use trigonometric functions:

• Let x = 2cos(t)
• Let y = 2sin(t)

Here, t is a parameter that varies, describing the angle around the circular cross-section of the cylinder. The values of t range from 0 to 2π.

Now we can substitute these expressions for x and y into the equation for z:

z = xy = (2cos(t))(2sin(t)) = 4cos(t)sin(t)

Using the double angle identity, we can further simplify this:

z = 2sin(2t)

Putting it all together, we can express the curve of intersection as a vector function:

Vector Function:

r(t) = [2cos(t), 2sin(t), 2sin(2t)] where t is in the interval [0, 2π].

This vector function r(t) represents the curve of intersection of the cylinder and the surface in three-dimensional space.