Chapter 7: Triple Integration
Section 7.5: Spherical Coordinates
A region R′ in the spherical coordinates ρ,φ,θ is mapped between a region R in Cartesian coordinates by the equations in Table 7.5.1.
Spherical to Cartesian
Cartesian to Spherical
Table 7.5.1 Equations for spherical coordinates
the equations on the left mapping R′ to R; those on the right, R to R′. Figure 7.5.1 defines the spherical coordinate system typically used in mathematics texts. (Be aware that physics and engineering texts reverse the names θ and φ.) Figure 7.5.2 shows three coordinate surfaces: the red surface is ρ=1/2; the blue, φ=π/3; and the green, θ=π/3.
Figure 7.5.1 Spherical coordinates (math texts)
Figure 7.5.2 Coordinate surfaces, spherical
The angle φ, called the polar angle, zenith angle, or colatitude, is measured downward from the positive z-axis, and satisfies the inequality 0≤φ≤π. (Some texts use the equation φ=arctanx2+y2/z, thereby expressing both angles as arctangents.)
The angle θ, called the azimuthal angle, is measured around the z-axis counterclockwise from the positive x-axis, and satisfies the inequality 0≤θ≤2 π, or its equivalent.
There is no uniformity in math or science texts about the order of the triple of coordinate names. There are about as many authors who write fρ,φ,θ as there are who write fρ,θ,φ. Since the meaning of the triple ρ,φ,θ depends on the text, it is imperative that users verify conventions in any text containing calculations in spherical coordinates.
The conventions in Maple are equally conflicted. For example, Table 7.5.2 illustrates the syntax by means of which the plot3d command will graph a surface defined in spherical coordinates.
plot3dfn1,n2,n1=range 1, n2=range 2, coords=spherical
plot3dρu,v,θu,v,φu,v,u=range u, v=range v,coords=spherical
Table 7.5.2 Syntax by which the plot3d command will graph in cylindrical coordinates
In the explicit case, the first range and its coordinate name is interpreted as the angle θ; the second, as φ. The function f is interpreted as ρφ,θ.
In the parametric case, the parameters u and v could be any two of ρ,φ,θ. For example, if u=θ and v=φ, then the parametric list would be ρφ,θ,θ,φ.
Note the order of the angles for the plot3d command; in computational commands, Maple uses the order φ, θ, but in graphing, the order is reversed! Hence, in the Student VectorCalculus package, the order of the coordinates is taken as ρ,φ,θ, and, regardless of the names used, the middle name must be that of the polar angle, that is, of the angle measured downward from the positive z-axis.
Both the Student packages VectorCalculus and MultivariateCalculus use r and not ρ as the default radial variable. However, both packages use r for polar and spherical coordinates. Since the radial variable of polar coordinates is not the radial variable of spherical coordinates, this manual insists on using different letters for these two coordinates. To use other than the default names in these packages, all the variable names must be explicitly given in a list following the system name.
Express the Cartesian point 2,5,3 in spherical coordinates.
Express the spherical point ρ,φ,θ=2,π/3,π/6 in Cartesian coordinates.
Sketch the region above the plane z=3/4 in a unit sphere with center at the origin.
Sketch the region above the cone z=x2+y2 but below the unit sphere that is centered at 0,0,1.
Sketch the region inside the unit sphere centered at the origin, outside the cone z=x2+y2, and above the plane z=0.
Sketch the region bounded below by the cone φ=π/6 and above by the sphere ρ=2.
Sketch the region between the spheres ρ=1 and ρ=2, and above the cone φ=π/3.
Sketch the region above the cone φ=π/4 and below the sphere ρ=3 cosφ.
Sketch the region enclosed by the surface ρ=1−cosφ.
Sketch the region bounded inside by the surface ρ=1+cosφ and outside by the sphere ρ=2.
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