Editing Integration by substitution (section)

== Substitution for multiple variables ==

One may also use substitution when integrating [[Multivariate function|functions of several variables]].

Here, the substitution function {{math|1=(''v''<sub>1</sub>,...,''v''<sub>''n''</sub>) = ''φ''(''u''<sub>1</sub>, ..., ''u''<sub>''n''</sub>)}} needs to be [[injective]] and continuously differentiable, and the differentials transform as:
<math display="block">dv_1 \cdots dv_n = \left|\det(D\varphi)(u_1, \ldots, u_n)\right| \, du_1 \cdots du_n,</math>
where {{math|det(''Dφ'')(''u''<sub>1</sub>, ..., ''u''<sub>''n''</sub>)}} denotes the [[determinant]] of the [[Jacobian matrix]] of [[partial derivative]]s of {{math|''φ''}} at the point {{math|(''u''<sub>1</sub>, ..., ''u''<sub>''n''</sub>)}}. This formula expresses the fact that the [[absolute value]] of the determinant of a matrix equals the volume of the [[Parallelepiped#Parallelotope|parallelotope]] spanned by its columns or rows.

More precisely, the ''[[change of variables]]'' formula is stated in the next theorem:

{{math theorem | math_statement = Let {{math|''U''}} be an open set in {{math|'''R'''<sup>''n''</sup>}} and {{math|''φ'' : ''U'' → '''R'''<sup>''n''</sup>}} an [[Injective function|injective]] differentiable function with continuous partial derivatives, the Jacobian of which is nonzero for every {{mvar|x}} in {{mvar|U}}.  Then for any real-valued, compactly supported, continuous function {{mvar|f}}, with support contained in {{math|''φ''(''U'')}}:
<math display="block">\int_{\varphi(U)} f(\mathbf{v})\, d\mathbf{v} = \int_U f(\varphi(\mathbf{u})) \,\,\left|\!\det(D\varphi)(\mathbf{u})\right| \,d\mathbf{u}.</math>
}}

The conditions on the theorem can be weakened in various ways. First, the requirement that {{mvar|φ}} be continuously differentiable can be replaced by the weaker assumption that {{mvar|φ}} be merely differentiable and have a continuous inverse.<ref>{{harvnb|Rudin|1987|loc=Theorem 7.26}}</ref> This is guaranteed to hold if {{mvar|φ}} is continuously differentiable by the [[inverse function theorem]].  Alternatively, the requirement that {{math|det(''Dφ'') ≠ 0}} can be eliminated by applying [[Sard's theorem]].<ref>{{harvnb|Spivak|1965|p=72}}</ref>

For Lebesgue measurable functions, the theorem can be stated in the following form:<ref>{{harvnb|Fremlin|2010|loc=Theorem 263D}}</ref>

{{math theorem | math_statement = Let {{mvar|U}} be a measurable subset of {{math|'''R'''<sup>''n''</sup>}} and {{math|''φ'' : ''U'' → '''R'''<sup>''n''</sup>}} an [[injective function]], and suppose for every {{mvar|x}} in {{mvar|U}} there exists {{math|''φ''&prime;(''x'')}} in {{math|'''R'''<sup>''n'',''n''</sup>}} such that {{math|1=''φ''(''y'') = ''φ''(''x'') + ''φ&prime;''(''x'')(''y'' − ''x'') + ''o''({{norm|''y'' − ''x''}})}} as {{math|''y'' → ''x''}} (here {{mvar|o}} is [[Landau symbol#Related asymptotic notations|little-''o'' notation]]). Then {{math|''φ''(''U'')}} is measurable, and for any real-valued function {{mvar|f}} defined on {{math|''φ''(''U'')}}:
<math display="block">\int_{\varphi(U)} f(v)\, dv = \int_U f(\varphi(u)) \,\,\left|\!\det \varphi'(u)\right| \,du</math>
in the sense that if either integral exists (including the possibility of being properly infinite), then so does the other one, and they have the same value.}}

Another very general version in [[measure theory]] is the following:<ref>{{harvnb|Hewitt|Stromberg|1965|loc=Theorem 20.3}}</ref>

{{math theorem | math_statement =  Let {{mvar|X}} be a [[locally compact]] [[Hausdorff space]] equipped with a finite [[Radon measure]] {{mvar|μ}}, and let {{mvar|Y}} be a [[Σ-compact space|σ-compact]] Hausdorff space with a [[sigma finite measure|&sigma;-finite]] Radon measure {{mvar|ρ}}.  Let {{math|''φ'' : ''X'' → ''Y''}} be an [[absolutely continuous]] function (where the latter means that {{math|1=''ρ''(''φ''(''E'')) = 0}} whenever {{math|1=''μ''(''E'') = 0}}). Then there exists a real-valued [[Borel algebra|Borel measurable function]] {{mvar|w}} on {{mvar|X}} such that for every [[Lebesgue integral|Lebesgue integrable]] function {{math|''f'' : ''Y'' → '''R'''}}, the function {{math|(''f'' ∘ ''φ'') ⋅ ''w''}} is Lebesgue integrable on {{mvar|X}}, and
<math display="block">\int_Y f(y)\,d\rho(y) = \int_X (f\circ \varphi)(x)\,w(x)\,d\mu(x).</math>
Furthermore, it is possible to write
<math display="block">w(x) = (g\circ \varphi)(x)</math>
for some Borel measurable function {{mvar|g}} on {{mvar|Y}}.}}

In [[geometric measure theory]], integration by substitution is used with [[Lipschitz function]]s. A bi-Lipschitz function is a Lipschitz function {{math|''φ'' : ''U'' → '''R'''<sup>n</sup>}} which is injective and whose inverse function {{math|''φ''<sup>&minus;1</sup> : ''φ''(''U'') → ''U''}} is also Lipschitz.  By [[Rademacher's theorem]], a bi-Lipschitz mapping is differentiable [[almost everywhere]]. In particular, the Jacobian determinant of a bi-Lipschitz mapping {{math|det ''Dφ''}} is well-defined almost everywhere.  The following result then holds:

{{math theorem | math_statement = Let {{mvar|U}} be an open subset of {{math|'''R'''<sup>n</sup>}} and {{math|''φ'' : ''U'' → '''R'''<sup>n</sup>}} be a bi-Lipschitz mapping.  Let {{math|''f'' : ''φ''(''U'') → '''R'''}} be measurable.  Then
<math display="block">\int_{\varphi(U)} f(x)\,dx = \int_U (f\circ \varphi)(x) \,\,\left|\!\det D\varphi(x)\right|\,dx</math>
in the sense that if either integral exists (or is properly infinite), then so does the other one, and they have the same value.}}

The above theorem was first proposed by [[Euler]] when he developed the notion of [[double integrals]] in 1769. Although generalized to triple integrals by [[Lagrange]] in 1773, and used by [[Adrien-Marie Legendre|Legendre]], [[Laplace]], and [[Gauss]], and first generalized to {{mvar|n}} variables by [[Mikhail Ostrogradsky]] in 1836, it resisted a fully rigorous formal proof for a surprisingly long time, and was first satisfactorily resolved 125 years later, by [[Élie Cartan]] in a series of papers beginning in the mid-1890s.<ref>{{harvnb|Katz|1982}}</ref><ref>{{harvnb|Ferzola|1994}}</ref>