I THINKKK To find a vector parametric equation for the part of the saddle z = xy inside the cylinder x^2 + y^2 = 9, we can use the parameterization:
x = r cos t
y = r sin t
z = r cos t sin t
where r is the radius of the cylinder and t is the angle of rotation around the z-axis.
Substituting the equation for z into the cylinder equation, we get:
x^2 + y^2 = 9 - z^2 = 9 - r^2 cos^2 t sin^2 t
Solving for r, we get:
r = sqrt(9 - x^2 - y^2)
Substituting r into the equation for z, we get:
z = xy = (r cos t)(r sin t) = r^2 sin t cos t
Therefore, the vector parametric equation for the part of the saddle z = xy inside the cylinder x^2 + y^2 = 9 is:
r(t) = sqrt(9 - x^2 - y^2)
x(t) = r cos t
y(t) = r sin t
z(t) = r^2 sin t cos t
where 0 <= t <= 2π.