Editing Distribution (mathematics) (section)

===Composition with a smooth function===

Let <math>T</math> be a distribution on <math>U.</math> Let <math>V</math> be an open set in <math>\R^n</math> and <math>F : V \to U.</math> If <math>F</math> is a [[Submersion (mathematics)|submersion]] then it is possible to define
<math display=block>T \circ F \in \mathcal{D}'(V).</math>

This is {{em|the '''composition''' of the distribution <math>T</math> with <math>F</math>}}, and is also called {{em|the '''[[Pullback (differential geometry)|pullback]]''' of <math>T</math> along <math>F</math>}}, sometimes written
<math display=block>F^\sharp : T \mapsto F^\sharp T = T \circ F.</math>

The pullback is often denoted <math>F^*,</math> although this notation should not be confused with the use of '*' to denote the adjoint of a linear mapping.

The condition that <math>F</math> be a submersion is equivalent to the requirement that the [[Jacobian matrix and determinant|Jacobian]] derivative <math>d F(x)</math> of <math>F</math> is a [[surjective]] linear map for every <math>x \in V.</math> A necessary (but not sufficient) condition for extending <math>F^{\#}</math> to distributions is that <math>F</math> be an [[open mapping]].<ref>See for example {{harvnb|Hörmander|1983|loc=Theorem 6.1.1}}.</ref> The [[Inverse function theorem]] ensures that a submersion satisfies this condition.

If <math>F</math> is a submersion, then <math>F^{\#}</math> is defined on distributions by finding the transpose map. The uniqueness of this extension is guaranteed since <math>F^{\#}</math> is a continuous linear operator on <math>\mathcal{D}(U).</math> Existence, however, requires using the [[Integration by substitution|change of variables]] formula, the inverse function theorem (locally), and a [[partition of unity]] argument.<ref>See {{harvnb|Hörmander|1983|loc=Theorem 6.1.2}}.</ref>

In the special case when <math>F</math> is a [[diffeomorphism]] from an open subset <math>V</math> of <math>\R^n</math> onto an open subset <math>U</math> of <math>\R^n</math> change of variables under the integral gives:
<math display=block>\int_V \phi\circ F(x) \psi(x)\,dx = \int_U \phi(x) \psi \left(F^{-1}(x) \right) \left|\det dF^{-1}(x) \right|\,dx.</math>

In this particular case, then, <math>F^{\#}</math> is defined by the transpose formula:
<math display=block>\left\langle F^\sharp T, \phi \right\rangle = \left\langle T, \left|\det d(F^{-1}) \right|\phi\circ F^{-1} \right\rangle.</math>