Question: Consider the curve (r = 4, $\theta$ = t, z = t) where this curve is written in cylindrical coordinates.

(a) What surface is this restricted to?

(b) Compute its tangent vector at t = $\pi$. First do this relative to r, $\theta$, z.

(c) Now convert the curve to one written in rectangular coordinates.

(d) Compute the tangent vector relative to x, y, z.

(e) Convert that tangent vector to r, 0, z coordinates.

(f) Do the two procedures agree on giving a tangent vector in cylindrical coordinates?

(g) If not, can you come up with a conversion procedure that does?