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?