changeofvariables

Change of variables is a technique in mathematics for rewriting problems in terms of new variables that simplify analysis or computation. It is central to the chain rule, to substitution methods in integration, and to the study of different coordinate systems.

In single-variable calculus, the substitution rule replaces a function of x with a function of a new variable u = g(x). If the substitution is differentiable and invertible, the differentials relate by du = g'(x) dx, so dx = du / g'(x). For definite integrals, the limits are transformed accordingly. The idea is to express the integral in terms of u and the corresponding differential, then integrate with respect to u.

In multiple dimensions, change of variables uses a differentiable mapping from the new variables to the original ones, usually written as φ: (u,v) -> (x,y). The area element transforms by the Jacobian determinant J = det ∂(x,y)/∂(u,v). The formula dx dy = |J| du dv leads to ∫∫_R f(x,y) dx dy = ∫∫_{R*} f(x(u,v), y(u,v)) |J| du dv, where R* is the region in the (u,v) space corresponding to R. A valid change of variables requires the Jacobian to be nonzero (local invertibility).

Common choices of new variables include polar, cylindrical, or spherical coordinates, chosen to exploit symmetry or to straighten curves and surfaces. Applications span physics, engineering, and probability, where densities, integrals, and geometric properties are transformed under coordinate changes. In differential geometry, change of variables is formalized via diffeomorphisms and the pullback of differential forms, ensuring integrals are preserved under smooth coordinate changes.