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 p (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 p (b) ~r(u; v) = hu; sin u cos v; sin u sin vi (e) ~r(u; v) = hu; v; u2 + v2i p p (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(jxj+π))p4 jx1337 − x42 + yj + 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.