Find parameterizations for the following objects. (Your parameterizations should, as always, be in rectangular coordinates, but thinking about cylindrical or spherical coordinates in the process might be easier.) (a) A curve: a quarter-circle from the point (0,0,1) to the point (½, 13,0) that lies on the sphere x2 y2 + 2 = 1. (b) A curve: the ellipse at the intersection of plane y + z = 0 and the cylinder x² + y² = 1. (c) A surface: the portion of the paraboloid z = x² + y² with 1 ≤ z ≤ 2. (d) The same surface as in (c), but flipped over and moved down so that what was once its top edge now rests on the xy-plane.

 Find parameterizations for the following objects. (Your parameterizations should, as always,
be in rectangular coordinates, but thinking about cylindrical or spherical coordinates in the process might be easier.)
(a) A curve: a quarter-circle from the point (0, 0, 1) to the point (1/2,√3/2, 0) that lies on the
sphere x^2 + y^2 + z^2 = 1.
(b) A curve: the ellipse at the intersection of plane y + z = 0 and the cylinder x^2 + y^2 = 1.
(c) A surface: the portion of the paraboloid z = x^2 + y^2 with 1 ≤ z ≤ 2.
(d) The same surface as in (c), but flipped over and moved down so that what was once its
top edge now rests on the xy-plane.