Math 21a Other coordinate systems and parametrized surfaces Fall 2017

Cylindrical coordinates In the cylindrical coordinate system, a point P in 3D is represented by a triple (r, θ, z) of numbers, where (r, θ) are the polar coordinates for the projection of P onto the xy-plane, and z is the directed distance from the xy-plane to P . To convert between cylindrical and Cartesian coordinates, we use the equations

x = r cos θ, y = r sin θ, z = z, p y r = x2 + y2, tan θ = , z = z. x

Spherical coordinates In the spherical coordinate system, a point P in 3D is represented by a triple (ρ, θ, φ) of numbers, where ρ ≥ 0 is the distance from the origin to P , θ is the same angle as in −→ cylindrical coordinates, and φ is the angle between the positive z-axis and the ray OP . We always take 0 ≤ φ ≤ π. To convert between spherical and Cartesian coordinates, we use the equations

x = ρ sin φ cos θ, y = ρ sin φ sin θ, z = ρ cos φ, p y z ρ = x2 + y2 + z2, tan θ = , cos φ = . x px2 + y2 + z2

For problems 1–3, give four descriptions of each geometric object: (a) An equation in Cartesian coordinates (the usual x, y, z) (b) An equation in cylindrical coordinates (r, θ, z) (c) An equation in spherical coordinates (ρ, θ, φ) (d) A description or sketch of the object (where appropriate).

1 Points Cartesian Cylindrical Spherical

√ (2, 2 3, 3)

π (3, 4 , 3)

π π (2, 3 , 6 ) 2 Surfaces Cartesian Cylindrical Spherical Description

π θ = 4

x2 +y2 +z2 −2z = 0

r = 2 cos θ

π φ = 4

z2 − x2 − y2 = 1

ρ2(sin2 φ sin2 θ + cos2 φ) = 9 3 Solids

Cartesian Cylindrical Spherical Description

3 ≤ r ≤ 5

π π 6 ≤ φ ≤ 4

cos θ ≤ r ≤ 1

A ball of radius 2 centered at the origin with a cylinder of radius 1 centered on the z-axis removed Parametrized surfaces 4 Match the following surfaces to their parametric equations.

I II III

IV V VI

(a) ~r(u, v) = hsin u cos v, cos u, sin u sin vi (d) ~r(u, v) = hu + 1, v − 2, 3 − u/2 − v/5i √ (b) ~r(u, v) = hu, sin u cos v, sin u sin vi (e) ~r(u, v) = hu, v, u2 + v2i √ √ (c) ~r(u, v) = hv cos u, 2v sin u, 2v2i (f) ~r(u, v) = h 1 + u2 cos v, 1 + u2 sin v, ui

5 Give a parametric equation for the plane containing the points (2, 2, 2), (−4, 0, 4) and (1, 3, 5).

6 Let f(x, y) = yesin(log log(|x|+π))p4 |x1337 − x42 + y| + 9001y. Find a parametric equation for the graph of f, i.e., the surface z = f(x, y). 7 Give a parametric equation for a sphere of radius 6 centered at (3, −1, −2). (Hint: use spherical coordinates, then modify the equation.)

8 Let x = ez + 1 be a curve in the xz-plane. Find a parametric equation for the surface obtained by revolving this parabola around the z-axis.

9 Find parametrizations for each of the following surfaces, given by an equation in cylin- drical or spherical coordinates. Then match each equation to its plot.

(a) r = 3 + 2 sin(z) cos(5θ)

(b) ρ = 6 + sin(7φ) sin(5θ)

(c) r = 4 + 2 sin(z + θ)

I II III 10 Here are some more surfaces for you to parametrize as practice.

(a)3 x + 2y + z = 6

(b) y = 2z2 − 3x2

(c) z = x2 + y2 + 1; what if we only want to parametrize the part of this elliptic paraboloid under the plane z = 10?

x2 y2 2 (d) 4 + 9 + z = 1

(e) The graph of the curve y = 1 + ex revolved around the x-axis

(f) The graph of the curve y = z2 + z revolved around the z-axis