1
Cylindrical and Spherical Coordinates
In 2 dimensions we have polar coordinates:
𝑥 = 𝑟 cos 𝜃
𝑦 = 𝑟 sin 𝜃 0 ≤ 𝜃 < 2𝜋
𝑟2 = 𝑥2 + 𝑦2
𝑦
tan 𝜃 = 𝑥
Cylindrical coordinates are just polar coordinates adding a 𝑧 coordinate.
𝑥 = 𝑟 cos 𝜃 𝑟2 = 𝑥2 + 𝑦2
𝑦
𝑦 = 𝑟 sin 𝜃 tan 𝜃 = 𝑥
𝑧=𝑧 𝑃(𝑟, 𝜃, 𝑧)
𝑧 𝑦
𝑟
𝜃
𝑥
𝜋
Ex. If (4, , 3) = (𝑟, 𝜃, 𝑧) are cylindrical coordinates, find the rectangular
6
coordinates of the same point.
𝜋
4 = 𝑟, = 𝜃, 3=𝑧.
6
𝜋 √3
𝑥 = 𝑟 cos 𝜃 = 4 cos 6 = 4 ( 2 ) = 2√3
𝜋 1
𝑦 = 𝑟 sin 𝜃 = 4 sin 6 = 4 (2) = 2
Rectangular coordinates: (𝑥, 𝑦, 𝑧) = (2√3, 2, 3).
Cylindrical and Spherical Coordinates
In 2 dimensions we have polar coordinates:
𝑥 = 𝑟 cos 𝜃
𝑦 = 𝑟 sin 𝜃 0 ≤ 𝜃 < 2𝜋
𝑟2 = 𝑥2 + 𝑦2
𝑦
tan 𝜃 = 𝑥
Cylindrical coordinates are just polar coordinates adding a 𝑧 coordinate.
𝑥 = 𝑟 cos 𝜃 𝑟2 = 𝑥2 + 𝑦2
𝑦
𝑦 = 𝑟 sin 𝜃 tan 𝜃 = 𝑥
𝑧=𝑧 𝑃(𝑟, 𝜃, 𝑧)
𝑧 𝑦
𝑟
𝜃
𝑥
𝜋
Ex. If (4, , 3) = (𝑟, 𝜃, 𝑧) are cylindrical coordinates, find the rectangular
6
coordinates of the same point.
𝜋
4 = 𝑟, = 𝜃, 3=𝑧.
6
𝜋 √3
𝑥 = 𝑟 cos 𝜃 = 4 cos 6 = 4 ( 2 ) = 2√3
𝜋 1
𝑦 = 𝑟 sin 𝜃 = 4 sin 6 = 4 (2) = 2
Rectangular coordinates: (𝑥, 𝑦, 𝑧) = (2√3, 2, 3).
