volume element

The volume element is the little box of space whose contributions you add up when you compute a multiple integral — the dV in a triple integral, or its two-dimensional cousin the area element dA in a double integral. It is the infinitesimal amount of volume (or area) sitting at a point, and when you change coordinates its form changes with them, because curved coordinate grids carve space into cells of unequal size. Getting the volume element right is the difference between a correct integral and a silently wrong one.

In Cartesian coordinates the volume element is the obvious dV = dx dy dz, a tiny rectangular box of equal-length sides. In any other coordinate system you multiply by the absolute value of the Jacobian, which equals the product of the scale factors: dV = h1 h2 h3 du1 du2 du3. This is why dV = r dr d-theta dz in cylindrical coordinates (the cell is a wedge whose width grows with r) and dV = rho^2 sin(phi) d-rho d-phi d-theta in spherical coordinates (cells near the poles and near the origin are tiny, those near the equator at large radius are large). The same logic in two dimensions gives the polar area element dA = r dr d-theta. The geometric picture is always the same: the volume element is the volume of the tiny parallelepiped that the coordinate grid lines bound at that point.

Everything a multiple integral computes — mass, charge, center of mass, moment of inertia, average value, probability — is literally a sum of (density times volume element) over a region, so the volume element is the unsung hero of every one of those formulas. The single most common and most damaging mistake in all of multivariable integration is changing coordinates but forgetting to change the volume element: writing dr d-theta instead of r dr d-theta, or d-rho d-phi d-theta instead of rho^2 sin(phi) d-rho d-phi d-theta. The factor is not decoration; it encodes the fact that physical space is being measured, not the coordinate grid.