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 \(R\) into subregions \(R_k\) 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: \[ \rho_1 \le \rho \le \rho_2 \quad \phi_1 \le \phi \le \phi_2 \quad \text{and} \quad \theta_1 \le \theta \le \theta_2 \] The coordinate center of the cell is at: \[ \bar{\rho}=\dfrac{\rho_1+\rho_2}{2} \quad \bar{\phi}=\dfrac{\phi_1+\phi_2}{2} \quad \text{and} \quad \bar{\theta}=\dfrac{\theta_1+\theta_2}{2} \] and the coordinate dimensions of the cell are: \[ \Delta\rho=\rho_2-\rho_1 \quad \Delta\phi=\phi_2-\phi_1 \quad \text{and} \quad \Delta\theta=\theta_2-\theta_1 \]

3dsphgrid

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

In the limit as \(\Delta\rho\), \(\Delta\phi\) and \(\Delta\theta\) get small, \(\Delta\rho\) becomes \(d\rho\), \(\Delta\phi\) becomes \(d\phi\), \(\Delta\theta\) becomes \(d\theta\) and \(\Delta V\) becomes \(dV\). Consequently:

The spherical differential of volume is: \[ dV=\rho^2\sin\phi\,d\rho\,d\phi\,d\theta \]

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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Supported in part by NSF Grant #1123255