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One of the methods for calculating integrals. It consists in representing an integral of an expression of the form <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051730/i0517301.png" /> by an integral of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051730/i0517302.png" />. For a [[Definite integral|definite integral]] the formula of integration by parts is
 
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051730/i0517303.png" /></td> <td valign="top" style="width:5%;text-align:right;">(1)</td></tr></table>
+
{{MSC|26A06}}
  
It is applicable under the assumptions that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051730/i0517304.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051730/i0517305.png" /> and their derivatives <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051730/i0517306.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051730/i0517307.png" /> are continuous on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051730/i0517308.png" />.
+
[[Category:Analysis]]
  
The analogue of (1) for an [[Indefinite integral|indefinite integral]] is
+
{{TEX|done}}
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051730/i0517309.png" /></td> <td valign="top" style="width:5%;text-align:right;">(2)</td></tr></table>
+
One of the methods for calculating integrals. Consider a continuous function $u:[a,b]\to
 +
\mathbb R$ and a continuously differentiable function $v:[a,b]\to \mathbb R$. If $U$ is
 +
a primitive of $u$, the integration by parts formula for the definite integral $\int_a^b
 +
u(x) v(x) dx$ is
 +
\begin{equation}\label{e:by_parts}
 +
\int_a^b u(x) v(x)\, dx = U(b) v (b) - U(a) v(a) - \int_a^b U(x) v' (x)\, dx\, .
 +
\end{equation}
 +
The formula is an easy consequence of the [[Fundamental theorem of calculus]] and of the
 +
the [[Leibniz rule]], according to which
 +
\[
 +
x\mapsto U (x) v (x) - \int_a^x U (t) v' (t)\, dt
 +
\]
 +
is a primitive of $uv$. The latter assertion is also called ''formula of integration by parts for indefinite integrals''.
  
The analogue of (1) for a [[Multiple integral|multiple integral]] is
+
The formula \eqref{e:by_parts} is still valid under the
 +
assumption that $u$ is Lebesgue integrable and $v$ is [[Absolute continuity|absolutely continuous]], replacing Riemann integrals with Lebesgue integrals.
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051730/i05173010.png" /></td> <td valign="top" style="width:5%;text-align:right;">(3)</td></tr></table>
+
In higher dimension the analogue of \eqref{e:by_parts} is a consequence of the
 
+
[[Ostrogradski formula|Gauss formula]]. If $\Omega\subset {\mathbb R}^n$ is a bounded open set with $C^1$ boundary and $\nu$ denotes the outward unit normal to $\partial \Omega$, then
Here <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051730/i05173011.png" /> is a domain in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051730/i05173012.png" /> with smooth (or at least piecewise-smooth) boundary <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051730/i05173013.png" />; <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051730/i05173014.png" />; and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051730/i05173015.png" /> is the angle between the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051730/i05173016.png" />-axis and the outward normal to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051730/i05173017.png" />. Formula (3) holds if, e.g., <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051730/i05173018.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051730/i05173019.png" /> and their first-order partial derivatives are continuous on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051730/i05173020.png" />. If the integrals in (3) are understood as Lebesgue integrals, then the formula is true if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051730/i05173021.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051730/i05173022.png" /> belong to a [[Sobolev space|Sobolev space]]: <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051730/i05173023.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051730/i05173024.png" /> for any <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051730/i05173025.png" /> with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051730/i05173026.png" />.
+
the following formula holds for every pair of $C^1$ functions $u$ and $v$:
 
+
\[
====References====
+
\int_\Omega u \frac{\partial u}{\partial x_i} = \int_{\partial \Omega}
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  V.A. Il'in,  E.G. Poznyak,  "Fundamentals of mathematical analysis" , '''1–2''' , MIR  (1982)  (Translated from Russian)  {{MR|}} {{ZBL|}} </TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  L.D. Kudryavtsev,  "Mathematical analysis" , '''1–2''' , Moscow  (1970)  (In Russian)  {{MR|1617334}} {{MR|1070567}} {{MR|1070566}} {{MR|1070565}} {{MR|0866891}} {{MR|0767983}} {{MR|0767982}} {{MR|0628614}} {{MR|0619214}} {{ZBL|1080.00002}} {{ZBL|1080.00001}} {{ZBL|1060.26002}} {{ZBL|0869.00003}} {{ZBL|0696.26002}} {{ZBL|0703.26001}} {{ZBL|0609.00001}} {{ZBL|0632.26001}} {{ZBL|0485.26002}} {{ZBL|0485.26001}} </TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  S.M. Nikol'skii,   "A course of mathematical analysis" , '''1–2''' , MIR  (1977)  (Translated from Russian)  {{MR|}} {{ZBL|0397.00003}} {{ZBL|0384.00004}} </TD></TR></table>
+
uv\,  \nu_i - \int_\Omega u \frac{\partial v}{\partial x_i}
 
+
\]
 
+
($\nu_i$ denotes the $i$-th component of the vector $\nu$; moreover the functions $u$, $v$ and their partial derivatives are assumed to have continuous extensions up to the boundary). The formula is still valid if $u$ and
 
+
$v$ belong to the [[Sobolev space|Sobolev spaces]] $W^{1,q}$ and $W^{1,p}$ for exponents $p,q$ with
====Comments====
+
\[
Formula (1) is valid whenever both <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051730/i05173027.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051730/i05173028.png" /> are absolutely continuous (cf. [[Absolute continuity|Absolute continuity]], 3)) on the closed interval <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051730/i05173029.png" />. In this generality the integral must be taken in the Lebesgue sense (cf. [[Lebesgue integral|Lebesgue integral]]).
+
\frac{1}{p}+\frac{1}{q} \leq 1 + \frac{1}{n}\, .
 
+
\]
For additional references see also [[Improper integral|Improper integral]].
+
The assumptions on the regularity of $\partial \Omega$ can also be weakened (for instance the formula still holds for Lipschitz domains).
  
 
====References====
 
====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top"> E. HewittK.R. Stromberg,  "Real and abstract analysis" , Springer (1965)  {{MR|0188387}} {{ZBL|0137.03202}} </TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top"> K.R. Stromberg,  "Introduction to classical real analysis" , Wadsworth  (1981) {{MR|0604364}} {{ZBL|0454.26001}} </TD></TR></table>
+
{|
 +
|-
 +
|valign="top"|{{Ref|Ap}}||valign="top"|  T.M. Apostol,  "Mathematical analysis". Second edition. Addison-Wesley  (1974) {{MR|0344384}} {{ZBL|0309.2600}}
 +
|-
 +
|valign="top"|{{Ref|EG}}|| L.C. Evans, R.F. Gariepy, "Measure theory and fine properties of functions" Studies in Advanced Mathematics. CRC Press, Boca Raton, FL,  1992. {{MR|1158660}} {{ZBL|0804.2800}}
 +
|-
 +
|valign="top"|{{Ref|IlPo}}||valign="top"|  V.A. Il'in,  E.G. Poznyak,  "Fundamentals of mathematical analysis" '''1–2''' , MIR  (1982)  (Translated from Russian) {{MR|0687827}}    {{ZBL|0138.2730}}
 +
|-
 +
|valign="top"|{{Ref|Ku}}||valign="top"|  L.D. Kudryavtsev,  "Mathematical analysis" , '''1''' , Moscow (1973)    (In Russian) {{MR|0619214}} {{ZBL|0703.26001}}
 +
|-
 +
|valign="top"|{{Ref|Ni}}||valign="top"|  S.M. Nikol'skii,  "A course of mathematical analysis" , '''1''' , MIR    (1977(Translated from Russian) {{MR|0466435}} {{ZBL|0384.00004}}
 +
|-
 +
|valign="top"|{{Ref|Ru}}||valign="top"|  W. Rudin, "Principles of mathematical analysis", Third edition,  McGraw-Hill (1976) {{MR|038502}} {{ZBL|0346.2600}}  
 +
|-
 +
|valign="top"|{{Ref|Ru}}||valign="top"| K.R. Stromberg,  "Introduction to classical real analysis" , Wadsworth  (1981)
 +
|-
 +
|}

Latest revision as of 09:25, 16 August 2013

2010 Mathematics Subject Classification: Primary: 26A06 [MSN][ZBL]

One of the methods for calculating integrals. Consider a continuous function $u:[a,b]\to \mathbb R$ and a continuously differentiable function $v:[a,b]\to \mathbb R$. If $U$ is a primitive of $u$, the integration by parts formula for the definite integral $\int_a^b u(x) v(x) dx$ is \begin{equation}\label{e:by_parts} \int_a^b u(x) v(x)\, dx = U(b) v (b) - U(a) v(a) - \int_a^b U(x) v' (x)\, dx\, . \end{equation} The formula is an easy consequence of the Fundamental theorem of calculus and of the the Leibniz rule, according to which \[ x\mapsto U (x) v (x) - \int_a^x U (t) v' (t)\, dt \] is a primitive of $uv$. The latter assertion is also called formula of integration by parts for indefinite integrals.

The formula \eqref{e:by_parts} is still valid under the assumption that $u$ is Lebesgue integrable and $v$ is absolutely continuous, replacing Riemann integrals with Lebesgue integrals.

In higher dimension the analogue of \eqref{e:by_parts} is a consequence of the Gauss formula. If $\Omega\subset {\mathbb R}^n$ is a bounded open set with $C^1$ boundary and $\nu$ denotes the outward unit normal to $\partial \Omega$, then the following formula holds for every pair of $C^1$ functions $u$ and $v$: \[ \int_\Omega u \frac{\partial u}{\partial x_i} = \int_{\partial \Omega} uv\, \nu_i - \int_\Omega u \frac{\partial v}{\partial x_i} \] ($\nu_i$ denotes the $i$-th component of the vector $\nu$; moreover the functions $u$, $v$ and their partial derivatives are assumed to have continuous extensions up to the boundary). The formula is still valid if $u$ and $v$ belong to the Sobolev spaces $W^{1,q}$ and $W^{1,p}$ for exponents $p,q$ with \[ \frac{1}{p}+\frac{1}{q} \leq 1 + \frac{1}{n}\, . \] The assumptions on the regularity of $\partial \Omega$ can also be weakened (for instance the formula still holds for Lipschitz domains).

References

[Ap] T.M. Apostol, "Mathematical analysis". Second edition. Addison-Wesley (1974) MR0344384 Zbl 0309.2600
[EG] L.C. Evans, R.F. Gariepy, "Measure theory and fine properties of functions" Studies in Advanced Mathematics. CRC Press, Boca Raton, FL, 1992. MR1158660 Zbl 0804.2800
[IlPo] V.A. Il'in, E.G. Poznyak, "Fundamentals of mathematical analysis" , 1–2 , MIR (1982) (Translated from Russian) MR0687827 Zbl 0138.2730
[Ku] L.D. Kudryavtsev, "Mathematical analysis" , 1 , Moscow (1973) (In Russian) MR0619214 Zbl 0703.26001
[Ni] S.M. Nikol'skii, "A course of mathematical analysis" , 1 , MIR (1977) (Translated from Russian) MR0466435 Zbl 0384.00004
[Ru] W. Rudin, "Principles of mathematical analysis", Third edition, McGraw-Hill (1976) MR038502 Zbl 0346.2600
[Ru] K.R. Stromberg, "Introduction to classical real analysis" , Wadsworth (1981)
How to Cite This Entry:
Integration by parts. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Integration_by_parts&oldid=28220
This article was adapted from an original article by V.A. Il'inT.P. Lukashenko (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article