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| − | ''change of variable in an integral''
| + | [[Category:Analysis]] |
| | + | {{MSC|26A06}} |
| | + | {{TEX|done}} |
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| − | One of the methods for calculating an integral. It consists in transforming the integral by transition to another variable of integration. For the [[Definite integral|definite integral]] of a function of one variable the formula is | + | One of the methods for calculating an integral in one real variable. It consists in transforming the integral by transition to another variable of integration. For the [[Definite integral|definite integral]] the formula is |
| | + | \begin{equation}\label{e:change_of_var} |
| | + | \int_a^b f(x)\, dx = \int_{\alpha}^\beta f (\phi (x)) \phi' (x)\, dx\, . |
| | + | \end{equation} |
| | + | This formula holds, for instance, under the following assumptions: |
| | | | |
| − | <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/i051740/i0517401.png" /></td> <td valign="top" style="width:5%;text-align:right;">(1)</td></tr></table>
| + | * $f:[a,b]\to \mathbb R$ is continuous; |
| | + | * $\phi: [\alpha, \beta]\to [a,b]$ is continuously differentiable; |
| | + | * $\phi (\alpha) = a$ and $\phi (\beta)=b$. |
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| − | It is true under the assumptions: <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051740/i0517402.png" /> is continuous on the interval <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051740/i0517403.png" />, which is the range of a function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051740/i0517404.png" /> that is defined and continuous, together with its first derivative <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051740/i0517405.png" />, on an interval <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051740/i0517406.png" />, and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051740/i0517407.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051740/i0517408.png" />.
| + | However, these assumptions can be relaxed considerably: we refer to {{Cite|S}} and to [[Area formula]]. |
| | | | |
| − | The analogue of (1) for the [[Indefinite integral|indefinite integral]] is | + | The analogue of \eqref{e:change_of_var} for the [[Indefinite integral|indefinite integral]] is the assertion that, if $F$ is a primitive of $f$, then $F\circ \phi$ is a primitive of $(f\circ \phi) \phi'$, which is an obvious corollary of the chain rule. |
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| − | <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/i051740/i0517409.png" /></td> <td valign="top" style="width:5%;text-align:right;">(2)</td></tr></table>
| + | The formula \eqref{e:change_of_var} can be generalized to integrals in more than one variable: we refer to [[Change of variables in an integral]] and [[Area formula]]. |
| − | | |
| − | If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051740/i05174010.png" /> is defined and differentiable on some segment <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051740/i05174011.png" />, while <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051740/i05174012.png" /> has a primitive on the range of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051740/i05174013.png" />, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051740/i05174014.png" /> also has a primitive on the given segment, and (2) holds.
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| − | In the case of a multiple Riemann integral over a bounded closed <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051740/i05174015.png" />-dimensional measurable region <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051740/i05174016.png" /> (cf. [[Multiple integral|Multiple integral]]), the analogue of (1) is
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| − | | |
| − | <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/i051740/i05174017.png" /></td> <td valign="top" style="width:5%;text-align:right;">(3)</td></tr></table>
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| − | | |
| − | Formula (3) holds under the following assumptions: the function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051740/i05174018.png" /> is continuous in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051740/i05174019.png" />; the transformation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051740/i05174020.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051740/i05174021.png" />, maps a region <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051740/i05174022.png" /> in the space of variables <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051740/i05174023.png" /> one-to-one onto <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051740/i05174024.png" />; the functions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051740/i05174025.png" /> have in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051740/i05174026.png" /> continuous first-order partial derivatives, and their [[Jacobian|Jacobian]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051740/i05174027.png" /> does not vanish. Formula (3) holds under more general assumptions as well (it is not necessary to require that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051740/i05174028.png" /> be continuous in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051740/i05174029.png" />, and the Jacobian may vanish on a set of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051740/i05174030.png" />-dimensional measure zero).
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| − | | |
| − | ====References====
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| − | <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)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top"> L.D. Kudryavtsev, "Mathematical analysis" , '''1–2''' , Moscow (1970) (In Russian)</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)</TD></TR></table>
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| − | ====Comments====
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| − | For very general hypothesis under which (1) and (3) hold for Lebesgue integrals (cf. [[Lebesgue integral|Lebesgue integral]]) see [[#References|[a1]]].
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| − | | |
| − | For additional references see also [[Improper integral|Improper integral]].
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| | | | |
| | ====References==== | | ====References==== |
| − | <table><TR><TD valign="top">[a1]</TD> <TD valign="top"> K.R. Stromberg, "Introduction to classical real analysis" , Wadsworth (1981)</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|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) |
| | + | |- |
| | + | |} |
2020 Mathematics Subject Classification: Primary: 26A06 [MSN][ZBL]
One of the methods for calculating an integral in one real variable. It consists in transforming the integral by transition to another variable of integration. For the definite integral the formula is
\begin{equation}\label{e:change_of_var}
\int_a^b f(x)\, dx = \int_{\alpha}^\beta f (\phi (x)) \phi' (x)\, dx\, .
\end{equation}
This formula holds, for instance, under the following assumptions:
- $f:[a,b]\to \mathbb R$ is continuous;
- $\phi: [\alpha, \beta]\to [a,b]$ is continuously differentiable;
- $\phi (\alpha) = a$ and $\phi (\beta)=b$.
However, these assumptions can be relaxed considerably: we refer to [S] and to Area formula.
The analogue of \eqref{e:change_of_var} for the indefinite integral is the assertion that, if $F$ is a primitive of $f$, then $F\circ \phi$ is a primitive of $(f\circ \phi) \phi'$, which is an obvious corollary of the chain rule.
The formula \eqref{e:change_of_var} can be generalized to integrals in more than one variable: we refer to Change of variables in an integral and Area formula.
References
| [Ap] |
T.M. Apostol, "Mathematical analysis". Second edition. Addison-Wesley (1974) MR0344384 Zbl 0309.2600
|
| [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)
|