Difference between revisions of "Integration by parts"
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| − | + | {{MSC|26A06}} | |
| − | + | [[Category:Analysis]] | |
| − | + | {{TEX|done}} | |
| − | + | 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 | + | 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. | ||
| − | + | 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 | |
| − | + | 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 space|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==== | ====References==== | ||
| − | + | {| | |
| + | |- | ||
| + | |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
2020 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=13008
Integration by parts. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Integration_by_parts&oldid=13008
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