31.4 Determining the Area or Volume Element

Thus far we have addressed questions that arise in doing a flux integral.

In the case of a volume integral, the issues discussed up to now do not exist.

The integrand is the integrand itself, which is a scalar field, and the volume element is dxdydz.

However the issue of how to express this volume element when you change variables does arise, and we consider that question here. The question and its answer are quite similar in any dimension.

We note at this point that area integrals are easy special cases of flux integrals. If we want to integrate f(x, y)dA over an area A in the xy plane, we can invent a third dimension in the k direction, and imagine we are integrating the flux of the vector f(x, y)k through the surface A in that plane.

This would be the integral of (f(x, y)k)ndS , and with n = k in the xy plane, this flux integral becomes the integral of f dxdy. In this case as in the volume case, the problem of reducing the surface integral to integrals dxdy does not exist.

Suppose now we have a volume or flux integral and wish to change variables from x, y (and z if a volume integral) to say, u, v (and w if same).

We do this when we believe that doing so makes the integrand easier to handle, or sometimes if it simplifies the limits of integration.

We wish to determine how to express dxdy or dxdydz in terms of dudv or dudvdw.
We have already noted how to do this but we repeat it here because of its importance.

Making a small change in u induces changes in x, y and z that can be described by the equation

with similar expressions for the changes in position dP_v and dP_w caused by changes in v and w.

The volume in xyz space induced by small changes du, dv and dw is the volume of the parallepiped with sides dP_u, dP_v and dP_w, which is the magnitude of the determinant whose rows or columns are these vectors.

We can factor dudvdw out of the vectors and obtain that dxdydz = Jdudvdw where J is the magnitude of the determinant formed by the derivatives of x, y and z with respect to u, v and w.

In the case of area, the comparable expression is the same thing: the ratio of the area element in one set of variables dxdy to the other, dudv, is the magnitude of the determinant formed by the derivatives of x and y with respect to u and v. In each case the magnitude, also called the absolute value, of this determinant is called the Jacobian of this change of variables.