The Cartesian coordinate system provides a straightforward way to describe the location of points in space. Some surfaces, however, can be difficult to model with equations based on the Cartesian system. This is a familiar problem; recall that in two dimensions, polar coordinates often provide a useful alternative system for describing the location of a point in the plane, particularly in cases involving circles. In this section, we look at two different ways of describing the location of points in space, both of them based on extensions of polar coordinates. As the name suggests, cylindrical coordinates are useful for dealing with problems involving cylinders, such as calculating the volume of a round water tank or the amount of oil flowing through a pipe. Similarly, spherical coordinates are useful for dealing with problems involving spheres, such as finding the volume of domed structures. Spherical CoordinatesIn the Cartesian coordinate system, the location of a point in space is described using an ordered triple in which each coordinate represents a distance.
In the cylindrical coordinate system, location of a point in space is described using two distances ((r) and (z)) and an angle measure ((θ)). In the spherical coordinate system, we again use an ordered triple to describe the location of a point in space.
In this case, the triple describes one distance and two angles. Spherical coordinates make it simple to describe a sphere, just as cylindrical coordinates make it easy to describe a cylinder. Grid lines for spherical coordinates are based on angle measures, like those for polar coordinates. The LibreTexts libraries are and are supported by the Department of Education Open Textbook Pilot Project, the UC Davis Office of the Provost, the UC Davis Library, the California State University Affordable Learning Solutions Program, and Merlot. We also acknowledge previous National Science Foundation support under grant numbers 1246120, 1525057, and 1413739. Unless otherwise noted, LibreTexts content is licensed.
1.3 Maxwell’s Equation No. 1; Volume Integral. Gauss’ electrostatics law is also written as a volume integral: This equation states that the charge enclosed in a volume is equal to the volume charge density, r, (rho) summed for the entire volume. Q is the charge enclosed in the volume. R is the volume charge density in coulombs per cubic meter. The Laplace equation is also a special case of the Helmholtz equation. The general theory of solutions to Laplace's equation is known as potential theory. The solutions of Laplace's equation are the harmonic functions, which are important in branches of physics, notably.
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