Multidimensional Integration 5 | Change of Variables Formula [dark version]

TL;DR

The n-dimensional change of variables formula preserves integral values when switching coordinates via a C^1 diffeomorphism F between open sets in R^n.

Briefing Cornell Notes

Briefing

Multidimensional integration gets a powerful shortcut through the change of variables formula: by switching from coordinates x to new coordinates x̃ via a smooth, invertible map, the value of an n-dimensional integral can be preserved—even though the integrand and the region both transform. The payoff is practical: the “right” coordinate system can turn a hard integral into a straightforward one, because either the region becomes simpler or the transformed integrand becomes easier to integrate.

Start with a measurable function f defined on an open set U in R^n, and consider the integral over U of f(x) with respect to n-dimensional volume (written as ∫_U f(x) d^n x). The change of variables idea introduces a second open set Ũ in R^n and a coordinate transformation F: U → Ũ. For the formula to work cleanly, F must be a C^1 diffeomorphism: it is continuously differentiable, bijective, and its inverse F^{-1} is also continuously differentiable.

Under this setup, the substitution replaces x by x̃ = F(x) inside the integrand. The remaining adjustment comes from how volume elements scale under the transformation. In one dimension, the mnemonic is that dx becomes f’(x) dx̃ (or equivalently, the derivative factor appears). In higher dimensions, the scaling factor is not the derivative but the determinant of the Jacobian matrix of F. The Jacobian J_F(x) is an n×n matrix of partial derivatives, but the integral needs only det(J_F(x)). Because integrals in R^n do not assume an orientation, the formula uses the absolute value |det(J_F(x))|.

The resulting n-dimensional change of variables formula states that integrating f over U can be done equivalently by integrating a transformed function over Ũ. Concretely, the transformed integral uses f(F^{-1}(x̃)) and multiplies by |det(J_{F^{-1}}(x̃))| (depending on which direction the substitution is written). The key structural rule is: inside f, substitute the corresponding expression for the old variables in terms of the new ones; outside, multiply by the absolute determinant of the relevant Jacobian.

A central consequence is symmetry: if one of the integrals exists, the other exists too, so the transformation does not break integrability. That means the formula can be applied in whichever direction makes the problem easier. Even if the expression on the transformed side looks more complicated, the combination of the original function f and the Jacobian factor can simplify dramatically for the right choice of F. The real challenge in applications is therefore not the formula itself, but selecting a suitable transformation F—often chosen so that the region or integrand becomes manageable. Examples are promised for the next installment.

Cornell Notes

The change of variables formula in R^n lets you compute ∫_U f(x) d^n x by switching to new coordinates x̃ = F(x), where F is a C^1 diffeomorphism between open sets U and Ũ. The integrand transforms by substituting the old variables with the corresponding expression in terms of the new variables (using F^{-1} when needed). The volume element d^n x changes by a factor of |det(J_F)|, the absolute value of the determinant of the Jacobian matrix. Because the formula preserves integral values (and integrability), it can be applied in either direction—whichever side yields a simpler region or integrand.

What conditions must the coordinate transformation F satisfy for the n-dimensional change of variables formula to apply?

F must be a C^1 diffeomorphism between open sets: it maps U ⊂ R^n to Ũ ⊂ R^n, is continuously differentiable, is bijective, and has a continuously differentiable inverse F^{-1}. This ensures the Jacobian determinant is well-defined and the substitution works reliably.

Why does the Jacobian determinant appear, and why is it taken in absolute value?

The Jacobian matrix J_F(x) captures how the transformation stretches and shears space locally. The n-dimensional volume element scales by det(J_F(x)), so the integral needs that factor. Since integrals in R^n do not depend on orientation, the formula uses |det(J_F(x))| to avoid sign changes from orientation flips.

How does the integrand change when switching from x to x̃?

The integrand f(x) is rewritten in terms of the new variables. If x̃ = F(x), then x = F^{-1}(x̃), so the transformed integrand becomes f(F^{-1}(x̃)). The exact placement of F or F^{-1} depends on whether the substitution is written from U to Ũ or from Ũ back to U.

What is the practical benefit of the formula beyond algebraic correctness?

The formula preserves the integral value while allowing a coordinate system change that can simplify the problem. Often the transformed region Ũ is easier to describe, or the product of the transformed integrand with the Jacobian factor becomes simpler to integrate. The main task becomes choosing a transformation F that makes the resulting expression manageable.

What does the existence claim mean for applications?

If one side of the change of variables formula defines a finite (or otherwise existing) integral, then the other side also exists. This lets practitioners confidently switch coordinates without worrying that the transformation will destroy integrability.

Review Questions

In the n-dimensional change of variables formula, what role does |det(J_F)| play, and how does it relate to the Jacobian matrix?
Why is F required to be a C^1 diffeomorphism rather than just differentiable?
When substituting x̃ = F(x), how do you determine whether the transformed integrand uses F or F^{-1}?

Key Points

1
The n-dimensional change of variables formula preserves integral values when switching coordinates via a C^1 diffeomorphism F between open sets in R^n.
2
The transformation must be continuously differentiable with a continuously differentiable inverse, ensuring a valid Jacobian-based scaling.
3
The integrand transforms by substituting old variables with expressions in the new variables, typically using F^{-1}(x̃) when x̃ = F(x).
4
The differential volume element d^n x scales by the absolute determinant of the Jacobian: |det(J_F)|.
5
Only the determinant of the Jacobian matters for the integral, not the full n×n matrix.
6
Integrability is preserved: if one integral exists, the other exists as well.
7
Choosing a good transformation F is the real challenge, since the Jacobian factor can combine with the transformed integrand to simplify the computation.

Highlights

A C^1 diffeomorphism F: U → Ũ enables coordinate changes in R^n without changing the value of an n-dimensional integral.

The Jacobian determinant enters as |det(J_F)| because it captures how n-dimensional volume scales under the transformation.

The formula can be applied in either direction, letting the problem be solved on whichever side is simpler.

The main difficulty in practice is selecting a transformation F that makes the transformed integrand and region easier to integrate.

Topics

Change of Variables
Jacobian Determinant
C1 Diffeomorphism
Multidimensional Integration
Coordinate Transformation