Distributions 9 | Coordinate Transformation [dark version]

TL;DR

Invertible coordinate transformations of distributions are defined through how distributions act on test functions, not by pointwise formulas.

Briefing Cornell Notes

Briefing

Invertible coordinate changes act on distributions by composing test functions with the inverse map and correcting by the Jacobian determinant. That rule is the bridge between ordinary change-of-variables for functions and the “right” definition for how generalized objects (distributions) transform—so that locally integrable functions stay locally integrable and non-regular distributions transform consistently.

Start with an invertible linear map a: ℝ^N → ℝ^N. Such a map can rotate, reflect, and stretch space, but it always sends lines to lines. The key invariance used early is that local integrability survives the coordinate change: if f is locally integrable, then f ∘ a is also locally integrable. The real question becomes how the corresponding distributions relate. For a distribution T, the transformation is defined by how T acts on test functions φ. One computes the transformed action by starting from the integral representation for regular distributions and applying the change-of-variables formula. The determinant of a enters through the absolute value |det(a)|, which adjusts the measure under the substitution y = a x. After substituting, the transformed integral can be rewritten as T applied to a specific transformed test function—namely φ composed with a^{-1}, scaled appropriately.

This yields the general definition: for an invertible linear map a, the transformed distribution T∘a (often written T after a) is defined so that (T after a)(φ) equals T applied to the test function obtained from φ by composing with a^{-1} and incorporating the Jacobian factor. The transcript emphasizes that this definition works for every distribution, not just regular ones, and it can be taken as the operational definition for non-regular distributions. A common “sloppy but readable” notation also appears: writing T(ax) as shorthand for the transformed distribution acting on φ(x) via the inverse substitution. Formally, it’s the same rule, just easier to read.

The same logic extends beyond linear maps. Adding a translation by a vector b leads to a transformation of the form T after (x ↦ a x + b), which corresponds to composing the test function with (x − b) under a^{-1}. More generally, any smooth coordinate transformation x ↦ g(x) can be used, provided g is C^∞ so that test functions remain compatible. In that case, the transformed distribution involves the Jacobian matrix of g, which may depend on x, and the determinant factor becomes the variable Jacobian correction.

A concrete payoff comes from the Delta distribution. Under a rotation a, the inverse equals the transpose and |det(a)| = 1, so the Jacobian correction disappears. Evaluating the transformed Delta on test functions reduces to checking the test function at zero, and because a^{-1}(0) = 0, the result is unchanged. The conclusion is that the Dirac delta distribution is invariant under rotations: Δ(x) = Δ(a x) for rotation matrices. The discussion closes by pointing toward the next topic—derivatives of distributions—framed as essential for further calculations.

Cornell Notes

Invertible coordinate transformations act on distributions by pushing forward the action through test functions. For an invertible linear map a, the transformed distribution is defined so that its value on a test function φ equals the original distribution’s value on a φ composed with a^{-1}, with a Jacobian determinant factor |det(a)| correcting the change of variables. This same principle extends to translations and, more generally, to smooth C^∞ maps using the Jacobian matrix of the transformation. A key example is the Dirac delta: under rotations, the determinant has absolute value 1 and the inverse sends 0 to 0, so Δ is unchanged. This invariance illustrates how properties known for functions carry over to distributions.

How does an invertible linear map a: ℝ^N → ℝ^N transform a distribution T?

The transformed distribution (often written T after a) is defined by how it acts on test functions φ. Using the change-of-variables formula, the action becomes an integral where the determinant correction |det(a)| appears. After substitution y = a x, the transformed expression can be rewritten as T applied to a test function built from φ composed with a^{-1} (and scaled by the Jacobian factor). In shorthand, the rule is: (T after a)(φ) = T( scaled-and-composed φ with a^{-1} ).

Why does the determinant |det(a)| matter in the transformation rule?

The determinant accounts for how volumes (and thus integration measures) change under the substitution y = a x. In the integral defining a regular distribution, the change-of-variables formula introduces the absolute determinant factor. The transcript notes that because a is invertible, det(a) ≠ 0, so dividing by |det(a)| is valid and the transformed integral matches the distribution defined via test functions.

How do translations affect distributions under x ↦ a x + b?

Translations shift the coordinates by b, so the test function must be shifted accordingly. The transformed distribution T after (x ↦ a x + b) corresponds to composing the test function with the inverse transformation, which becomes a^{-1}(x − b). The same Jacobian logic from the linear part applies, while the translation only changes where the test function is evaluated through the inverse shift.

What extra requirement appears when moving from linear maps to general coordinate transformations?

The transformation g must be smooth enough to preserve the structure of test functions: since test functions are C^∞, the map g must also be C^∞. The transformed distribution then uses the Jacobian matrix of g, whose determinant may depend on x. The determinant factor generalizes the constant |det(a)| from the linear case.

Why is the Dirac delta distribution invariant under rotations?

For a rotation a, the inverse equals the transpose and |det(a)| = 1, so the Jacobian correction becomes trivial. When applying the transformed delta to a test function φ, the delta evaluates the test function at zero. Since a^{-1}(0) = 0, the evaluation remains φ(0). Therefore Δ(x) and Δ(a x) act identically on every test function, meaning the distributions are the same.

Review Questions

Given an invertible linear map a, what two ingredients determine how (T after a) acts on a test function φ?
In the general smooth case x ↦ g(x), what role does the Jacobian matrix play in the transformed distribution?
What specific properties of a rotation matrix make the Dirac delta invariant under x ↦ a x?

Key Points

1
Invertible coordinate transformations of distributions are defined through how distributions act on test functions, not by pointwise formulas.
2
For an invertible linear map a, the Jacobian correction uses the absolute determinant |det(a)| to match the change-of-variables formula.
3
The transformed distribution for linear maps composes test functions with a^{-1}, with the determinant factor ensuring consistency of integrals.
4
Translations x ↦ a x + b shift the test function via the inverse map a^{-1}(x − b).
5
General smooth coordinate changes require a C^∞ transformation so test functions remain compatible.
6
Under rotations, the Dirac delta is invariant because |det(a)| = 1 and the inverse sends 0 to 0.

Highlights

The transformation rule for distributions mirrors the change-of-variables formula: compose test functions with the inverse map and correct by the Jacobian determinant.

A “sloppy” notation like T(ax) is justified as shorthand for the inverse-substitution rule on test functions.

The Dirac delta stays the same under rotations because zero is fixed and the determinant factor equals 1.

Smooth (nonlinear) coordinate transformations work as long as the map is C^∞, with the Jacobian determinant possibly depending on x.