Cylindrical and Spherical Coordinates
Section 15.7: Cylindrical and Spherical Coordinates
Goals: 1. To convert between rectangular, cylindrical, and spherical coordinates 2. To represent surfaces in space using cylindrical and spherical coordinates.
Problems that involve equations with either circular, cylindrical, or spherical symmetry are often difficult to work with in rectangular coordinates. For these types of problems we often use alternative coordinate systems.
The cylindrical coordinate system is simply an extension of the polar coordinate system that you learned about in previous classes. To create a system of cylindrical coordinates simply unite polar coordinates in the xy-plane with the regular z-axis to get an ordered triple θ zr ),,( . In other words, replace the x and y coordinates with the polar coordinates r and θ , leaving the z-coordinate unaltered (see the picture).
To convert from cylindrical to rectangular coordinates, use the following equations.
x = r cosθ , = ry sinθ , = zz
To convert from rectangular to cylindrical coordinates, use the following equations.
y r = + yx 222 , tanθ = , = zz x
Notes 1. Many equations in cylindrical coordinates are simpler than their counterparts in rectangular coordinates. In particular, equations that have symmetry with respect to the z-axis have simpler equations in cylindrical coordinates. For example, the rectangular equation for a cylinder =+ kyx 222 reduces to = kr in cylindrical coordinates. 2. The coordinate surfaces for the rectangular coordinate system are the planes perpendicular to the coordinate axes, x = a, y = b, and z = c. The coordinate surfaces for the cylindrical coordinate system are: r = a (a cylinder), θ = b (a plane perpendicular to the xy-plane containing the z-axis) and z = c (a plane perpendicular to the z-axis). Note that the system is named after the r = a surface.
The spherical coordinate system locates a point P in space using a distance and two angles, ρ θ ,,( φ) . Note that ρ (rho) is the distance between the origin and P. The angleθ is the same angle from cylindrical coordinates, and φ (phi) represents the angle between the positive z-axis and the line segment OP , 0.φ ≤≤ π
Note: You can also think of spherical coordinates as a system having two polar coordinates, the first residing the xy-plane, just as in cylindrical coordinates. The second angle is in the plane formed by rz (this is the r from cylindrical coordinates) with the z-axis as the polar axis.
To convert from spherical to rectangular coordinates, use the following equations.
x = ρ sinφ cosθ , y = ρ φ sinsin θ , z = ρ cosφ
To convert from rectangular to spherical coordinates, use the following equations.
y z ρ 2 = x ++ zy 222 , tanθ = , φ = cos−1 222 x ++ zyx
To convert from spherical to cylindrical coordinates, use the following equations.
r = ρ sinφ , θ = θ , z = ρ cosφ
To convert from cylindrical to spherical coordinates, use the following equations.
z ρ = r + z 22 , θ = θ , φ = cos−1 222 ++ zyx
Note: The coordinate surfaces in spherical coordinates are ρ = a (a sphere), θ = b (a half-plane perpendicular to the xy-plane containing the z-axis), and φ = c (a half-cone whose apex is at the origin and whose axis is along the z-axis.) The system is named after the ρ = a surface.