How to Integrate in a Spherical Coordinate System

Review of Integration Integration with Cartesian coordinates is simple. The general form is $\int\int\int f(x,y,z)dxdydz$ in which $f(x,y,z)$ is an arbitrary function of the Cartesian coordinates. However, there may be cases in which integrating with spherical coordinates is more convenient. Given the above, general form for integration with Cartesian coordinates, how can one integrate in a spherical coordinate system? How the Spherical Coordinate System is Different The first step is to convert $f(x,y,z)$ into a function $g(r,\phi,\theta)$ of the spherical coordinates, $r$, $\phi$, and $\theta$. Here, $\phi$ is the azimuthal angle in the $x-y$ plane, and $\theta$ is the polar angle. The above integral has three dummy variables: $dx$, $dy$, and $dz$. These dummy variables indicate that the summation for the integral should occur along the $x$, $y$, and $z$ axes, respectively. In the Cartesian coordinate system, these axes are mutually perpendicular, regardless of which points in space are being considered. In contrast, in spherical coordinates, the unit vectors $\vec{e}_r$, $\vec{e}_{\phi}$, and $\vec{e}_{\theta}$ change orientation depending on which point in space is considered. The unit vectors for the spherical coordinate system are not fixed like the unit vectors for the Cartesian coordinate system. In fact, even if $\vec{e}_r$, $\vec{e}_{\phi}$, and $\vec{e}_{\theta}$ are mutually perpendicular at a single point in space, once another point is considered, a new set of unit vectors $\vec{e}_r$, $\vec{e}_{\phi}$, and $\vec{e}_{\theta}$ is needed for that point in space. Given this realization, how can integration be done in spherical coordinates, if the unit vectors change for each point in space? This is certainly a complicated situation. To solve this problem, the process of integration needs to be closely examined. What is happening when one finds the area corresponding to an integral? This question is easier to answer one coordinate at a time. A Close Analysis of Integration So, I am going […]