p. 1/ Section 1.4: Cylindrical and Spherical Coordinates

p. 1/ Section 1.4: Cylindrical and Spherical Coordinates

p. / Cylindrical Coordinate (r,θ,w) where θ is measured counterclockwise as viewed from the positive w-axis.

p. / Cylindrical Coordinate (r,θ,w) where θ is measured counterclockwise as viewed from the positive w-axis. Rectangular and Cylindrical Coordinates x = r cosθ y = r sin θ z = w r = x + y tan θ = y x,x 0 w = z

p. / Cylindrical Coordinate (r,θ,w) where θ is measured counterclockwise as viewed from the positive w-axis. Rectangular and Cylindrical Coordinates x = r cosθ y = r sin θ z = w r = x + y tan θ = y x,x 0 w = z We assume r 0 and 0 θ < π except we explicitly relax it.

Cylindrical Coordinate (r,θ,w) where θ is measured counterclockwise as viewed from the positive w-axis. Rectangular and Cylindrical Coordinates x = r cosθ y = r sin θ z = w r = x + y tan θ = y x,x 0 w = z We assume r 0 and 0 θ < π except we explicitly relax it. Maple command: specify coords=cylindrical in plot3d. p. /

p. 3/ Examples for Cylindrical Coordinate What is rectangular of the point with cylindrical (7, 3π/4, )? What is cylindrical of the point with rectangular ( 7, 7, )? What is cylindrical of the point with rectangular ( 7, 7, )? What is cylindrical of the point with rectangular ( 7, 7, )?

p. 3/ Examples for Cylindrical Coordinate What is rectangular of the point with cylindrical (7, 3π/4, )? What is cylindrical of the point with rectangular ( 7, 7, )? What is cylindrical of the point with rectangular ( 7, 7, )? What is cylindrical of the point with rectangular ( 7, 7, )? Write the equations (a) x + y + (z 1) = 1 and (b) z = (x + y ) in cylindrical

p. 3/ Examples for Cylindrical Coordinate What is rectangular of the point with cylindrical (7, 3π/4, )? What is cylindrical of the point with rectangular ( 7, 7, )? What is cylindrical of the point with rectangular ( 7, 7, )? What is cylindrical of the point with rectangular ( 7, 7, )? Write the equations (a) x + y + (z 1) = 1 and (b) z = (x + y ) in cylindrical Sketch the graphs of the equations (in cylindrical ) (a) r = 5; (b) θ = 5π/6; (c) r = cos θ; (d) w = r; (e) w = r ; (f) w = r cos θ.

p. 4/ Spherical Coordinate (ρ,θ,φ)

p. 4/ Spherical Coordinate (ρ,θ,φ) Rectangular and Spherical Coordinates x = ρsin φ cos θ y = ρsin φ sin θ z = ρcos φ ρ = x + y + z tan θ = y x,x 0 z cos φ = x +y +z

p. 4/ Spherical Coordinate (ρ,θ,φ) Rectangular and Spherical Coordinates x = ρsin φ cos θ y = ρsin φ sin θ z = ρcos φ ρ = x + y + z tan θ = y x,x 0 z cos φ = x +y +z We assume ρ 0, 0 θ < π, and 0 φ π.

p. 4/ Spherical Coordinate (ρ,θ,φ) Rectangular and Spherical Coordinates x = ρsin φ cos θ y = ρsin φ sin θ z = ρcos φ ρ = x + y + z tan θ = y x,x 0 z cos φ = x +y +z We assume ρ 0, 0 θ < π, and 0 φ π. Maple command: specify coords=spherical in plot3d.

p. 5/ Examples for Spherical Coordinate Convert the spherical coordinate equations to Cartesian

p. 5/ Examples for Spherical Coordinate Convert the spherical coordinate equations to Cartesian 1. ρ = 5;

p. 5/ Examples for Spherical Coordinate Convert the spherical coordinate equations to Cartesian 1. ρ = 5;. φ = π/6;

p. 5/ Examples for Spherical Coordinate Convert the spherical coordinate equations to Cartesian 1. ρ = 5;. φ = π/6; 3. θ = π/4;

p. 5/ Examples for Spherical Coordinate Convert the spherical coordinate equations to Cartesian 1. ρ = 5;. φ = π/6; 3. θ = π/4; 4. ρ = 4 sec φ;

p. 5/ Examples for Spherical Coordinate Convert the spherical coordinate equations to Cartesian 1. ρ = 5;. φ = π/6; 3. θ = π/4; 4. ρ = 4 sec φ; 5. ρ = 4 cos φ;

p. 5/ Examples for Spherical Coordinate Convert the spherical coordinate equations to Cartesian 1. ρ = 5;. φ = π/6; 3. θ = π/4; 4. ρ = 4 sec φ; 5. ρ = 4 cos φ; Describe the points specified by the sets of spherical

p. 5/ Examples for Spherical Coordinate Convert the spherical coordinate equations to Cartesian 1. ρ = 5;. φ = π/6; 3. θ = π/4; 4. ρ = 4 sec φ; 5. ρ = 4 cos φ; Describe the points specified by the sets of spherical 1. {(ρ,θ,φ) ρ =,θ = π/4};

p. 5/ Examples for Spherical Coordinate Convert the spherical coordinate equations to Cartesian 1. ρ = 5;. φ = π/6; 3. θ = π/4; 4. ρ = 4 sec φ; 5. ρ = 4 cos φ; Describe the points specified by the sets of spherical 1. {(ρ,θ,φ) ρ =,θ = π/4};. {(ρ,θ,φ) φ = π/3,ρ 1}.