Question

Match the equations with the graphs labeled I–IV and give reasons for your answers. Determine which families of grid curves have constant and which have constant.

\(r(u,v) = sin\;v\,i + cos\;u\;sin\;2v\;j + sin\;u\;sin\;2v\;k\)

Short answer

The equation corresponds to image II

Step-by-step solution

Parametric equations

To determine which image is the image of the given surface in this case one can apply several strategies. The first thing to do is to write down the parametric equations of the surface. These are:

\(\begin{aligned}x(u,v) &= \sin v\\y(u,v) &= \cos u\sin 2v\\z(u,v) &= \sin u\sin 2v\end{aligned}\)

Explanation

Now, assume that value of v is fixed. For fixed v the coordinates xcoordinate becomes constant. For constant v equations\(y = \sin 2v\cos u\)and \(z = 2\sin 2v\sin u\)describe a circle of radius \(\sin 2u\).

This circle lies in the plane parallel to yz plane and the radii of the circles differ as we approach the yz plane.

The circle should vanish at the origin since that would mean that the value of x is equal to zero.

By taking a look at the given images, we see that the only surface that behaves on this way is the surface II. Hence the given equations correspond to the surface II.