Editing Integration by parts (section)

==Higher dimensions==
Integration by parts can be extended to functions of several variables by applying a version of the fundamental theorem of calculus to an appropriate product rule. There are several such pairings possible in multivariate calculus, involving a scalar-valued function ''u'' and vector-valued function (vector field) '''V'''.<ref>{{Cite web|url=http://www.math.nagoya-u.ac.jp/~richard/teaching/s2016/Ref2.pdf|title=The Calculus of Several Variables| last=Rogers| first=Robert C. |date=September 29, 2011}}</ref> 

The [[Vector calculus identities#First derivative identities|product rule for divergence]] states:

<math display="block">\nabla \cdot ( u \mathbf{V} ) \ =\ u\, \nabla \cdot \mathbf V \ +\  \nabla u\cdot \mathbf V.</math>

Suppose <math>\Omega</math> is an [[Open set|open]] [[bounded set|bounded subset]] of <math>\R^n</math> with a [[piecewise smooth]] [[boundary (topology)|boundary]] <math>\Gamma=\partial\Omega</math>. Integrating over <math>\Omega</math> with respect to the standard volume form <math>d\Omega</math>, and applying the [[divergence theorem]], gives:

<math display="block">\int_{\Gamma} u \mathbf{V} \cdot \hat{\mathbf n} \,d\Gamma \ =\ \int_\Omega\nabla\cdot ( u \mathbf{V} )\,d\Omega \ =\ \int_\Omega u\, \nabla \cdot \mathbf V\,d\Omega \ +\  \int_\Omega\nabla u\cdot \mathbf V\,d\Omega,</math>

where  <math>\hat{\mathbf n}</math> is the outward unit normal vector to the boundary, integrated with respect to its standard Riemannian volume form <math>d\Gamma</math>. Rearranging gives:

<math display="block">
\int_\Omega u \,\nabla \cdot \mathbf V\,d\Omega \ =\ \int_\Gamma u \mathbf V \cdot \hat{\mathbf n}\,d\Gamma - \int_\Omega  \nabla u \cdot \mathbf V \, d\Omega,
</math>

or in other words
<math display="block">
\int_\Omega u\,\operatorname{div}(\mathbf V)\,d\Omega  \ =\ \int_\Gamma u \mathbf V \cdot \hat{\mathbf n}\,d\Gamma - \int_\Omega  \operatorname{grad}(u)\cdot\mathbf V\,d\Omega   .
</math>
The [[Differentiability class|regularity]] requirements of the theorem can be relaxed. For instance, the boundary <math> \Gamma=\partial\Omega</math> need only be [[Lipschitz continuous]], and the functions ''u'', ''v''  need only lie in the [[Sobolev space]] <math>H^1(\Omega)</math>.

=== Green's first identity ===
Consider the continuously differentiable vector fields <math>\mathbf U = u_1\mathbf e_1+\cdots+u_n\mathbf e_n</math> and <math>v \mathbf e_1,\ldots, v\mathbf e_n</math>, where <math>\mathbf e_i</math> is the ''i''-th standard basis vector for <math>i=1,\ldots,n</math>. Now apply the above integration by parts to each <math>u_i</math> times the vector field <math>v\mathbf e_i</math>:

<math display="block">\int_\Omega u_i\frac{\partial v}{\partial x_i}\,d\Omega   \ =\ \int_\Gamma u_i v \,\mathbf e_i\cdot\hat\mathbf{n}\,d\Gamma - \int_\Omega \frac{\partial u_i}{\partial x_i} v\,d\Omega.</math>

Summing over ''i'' gives a new integration by parts formula:

<math display="block"> \int_\Omega \mathbf U \cdot \nabla v\,d\Omega \ =\ \int_\Gamma v \mathbf{U}\cdot \hat{\mathbf n}\,d\Gamma - \int_\Omega v\, \nabla \cdot \mathbf{U}\,d\Omega.</math>

The case <math>\mathbf{U}=\nabla u</math>, where <math>u\in C^2(\bar{\Omega})</math>, is known as the first of [[Green's identities]]:

<math display="block"> \int_\Omega \nabla u \cdot \nabla v\,d\Omega\ =\ \int_\Gamma v\, \nabla u\cdot\hat{\mathbf n}\,d\Gamma - \int_\Omega v\, \nabla^2 u \, d\Omega.</math>