cylindrical coordinates

Cylindrical coordinates are just polar coordinates in the plane with the ordinary height z stacked on top. A point in space is labeled (r, theta, z): r and theta describe where its shadow lands in the floor plane exactly as in polar coordinates, and z is how high it floats above that floor. Anything with an axis of symmetry — a pipe, a can, a tornado, a column of fluid swirling in a tank — is most comfortable in cylindrical coordinates, because the axis of the object becomes the z-axis and the round cross-section becomes a simple range of r and theta.

The conversion is x = r cos(theta), y = r sin(theta), z = z. Because the z-direction is left untouched, the Jacobian is exactly the polar one, r — so the volume element is dV = r dr d-theta dz. You can read this off geometrically: a little cylindrical box has a base of polar area r dr d-theta and a height dz, so its volume is the product. To integrate over a solid in cylindrical coordinates you typically let z run between a lower surface z = g(r, theta) and an upper surface z = h(r, theta), then sweep r and theta over the shadow of the solid in the plane — the same region-of-integration thinking as in two dimensions, with the extra height integral done first or last.

Cylindrical coordinates are the working tool for cylinders, cones, paraboloids of revolution, and any problem with rotational symmetry about an axis — flow in a pipe, the field of a charged wire, the moment of inertia of a flywheel, heat conduction in a rod. The Laplacian, the divergence, and the gradient all take standard forms in these coordinates that engineers use constantly. The one habit to keep: the r in r dr d-theta dz is the same forgettable, essential Jacobian factor as in plane polar coordinates — and theta is still cyclic, so watch for double-counting at theta = 0 and theta = 2pi being the same ray.