21. Multiple Integrals in Curvilinear Coordinates

c. Integrating in Spherical Coordinates

The extension of integration to spherical coordinates in 3D is somewhat similar to the extension to cylindrical coordinates. We break up the region of integration RR into subregions RkR_k which are spherical grid cells. We need to know the volume of these cells.

1. Grid Cells

A spherical box is very different from a cylindrical box or a polar rectangle. Think of a rectangle on the surface of the earth between two lines of lattitude and between two lines of longitude. Pile up dirt on this rectangle to some depth and that is a spherical box.

A spherical box or spherical grid cell is a region of the form: ρ1ρρ2ϕ1ϕϕ2andθ1θθ2 \rho_1 \le \rho \le \rho_2 \quad \phi_1 \le \phi \le \phi_2 \quad \text{and} \quad \theta_1 \le \theta \le \theta_2 The coordinate center of the cell is at: ρˉ=ρ1+ρ22ϕˉ=ϕ1+ϕ22andθˉ=θ1+θ22 \bar{\rho}=\dfrac{\rho_1+\rho_2}{2} \quad \bar{\phi}=\dfrac{\phi_1+\phi_2}{2} \quad \text{and} \quad \bar{\theta}=\dfrac{\theta_1+\theta_2}{2} and the coordinate dimensions of the cell are: Δρ=ρ2ρ1Δϕ=ϕ2ϕ1andΔθ=θ2θ1 \Delta\rho=\rho_2-\rho_1 \quad \Delta\phi=\phi_2-\phi_1 \quad \text{and} \quad \Delta\theta=\theta_2-\theta_1

3dsphgrid

It is difficult to derive an exact formula for the volume of a spherical box. However, it is approximately: ΔV=ρˉ2sinϕˉΔρΔϕΔθ \Delta V=\bar{\rho}\,^2\sin\bar{\phi}\,\Delta\rho\,\Delta\phi\,\Delta\theta

In the limit as Δρ\Delta\rho, Δϕ\Delta\phi and Δθ\Delta\theta get small, Δρ\Delta\rho becomes dρd\rho, Δϕ\Delta\phi becomes dϕd\phi, Δθ\Delta\theta becomes dθd\theta and ΔV\Delta V becomes dVdV. Consequently:

Differential of Volume in Spherical Coordinates
The spherical differential of volume is: dV=ρ2sinϕdρdϕdθ dV=\rho^2\sin\phi\,d\rho\,d\phi\,d\theta

Memorize this!

We will derive this formula for the spherical volume differential as a special case of the volume differential for general curvilinear coordinates in 3D later in this chapter. For now, we will just use the formula to do integrals starting on the next page.

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