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Differentiation under the integral sign : ウィキペディア英語版
Differentiation under the integral sign is a useful operation in calculus. Formally it can be stated as follows::Theorem. Let ''f''(''x'', ''t'') be a function such that both ''f''(''x'', ''t'') and its partial derivative ''fx''(''x'', ''t'') are continuous in ''t'' and ''x'' in some region of the (''x'', ''t'')-plane, including ''a''(''x'') ≤ ''t'' ≤ ''b''(''x''), ''x''0 ≤ ''x'' ≤ ''x''1. Also suppose that the functions ''a''(''x'') and ''b''(''x'') are both continuous and both have continuous derivatives for ''x''0 ≤ ''x'' ≤ ''x''1. Then for ''x''0 ≤ ''x'' ≤ ''x''1:::\fracx} \left (\int_^f(x,t)\,\mathrmt \right) = f(x,b(x))\cdot b'(x) - f(x,a(x))\cdot a'(x) + \int_^ \fracf(x,t)\; \mathrmt.This formula is the general form of the Leibniz integral rule and can be derived using the fundamental theorem of calculus. The (first) fundamental theorem of calculus is just a particular case of the above formula, for ''a''(''x'') = ''a'', a constant, ''b''(''x'') = ''x'' and ''f''(''x'', ''t'') = ''f''(''t'').If both upper and lower limits are taken as constants, then the formula takes the shape of an operator equation::''ItDx'' = ''DxIt'',where ''Dx'' is the partial derivative with respect to ''x'' and ''It'' is the integral operator with respect to ''t'' over a fixed interval. That is, it is related to the symmetry of second derivatives, but involving integrals as well as derivatives. This case is also known as the Leibniz integral rule.The following three basic theorems on the interchange of limits are essentially equivalent:* the interchange of a derivative and an integral (differentiation under the integral sign; i.e., Leibniz integral rule)* the change of order of partial derivatives* the change of order of integration (integration under the integral sign; i.e., Fubini's theorem)== Higher dimensions == The Leibniz integral rule can be extended to multidimensional integrals. In two and three dimensions, this rule is better known from the field of fluid dynamics as the Reynolds transport theorem::\fract} \int_ F(\vec, t) \,\mathrmV = \int_ \frac \,F(\vec, t)\,\mathrmV + \int_ \,F(\vec, t)\, \vec_b \cdot \mathrm\mathbfwhere F(\vec, t)\, is a scalar function, ''D''(''t'') and ∂''D''(''t'') denote a time-varying connected region of R3 and its boundary, respectively, \vec_b\, is the Eulerian velocity of the boundary (see Lagrangian and Eulerian coordinates) and dΣ = n dS is the unit normal component of the surface element.The general statement of the Leibniz integral rule requires concepts from differential geometry, specifically differential forms, exterior derivatives, wedge products and interior products. With those tools, the Leibniz integral rule in ''p''-dimensions is:,:\fract}\int_\omega=\int_ i__x\omega)+\int_ i_\dot,\,where Ω(''t'') is a time-varying domain of integration, ω is a ''p''-form, \vec\, is the vector field of the velocity, \vec=\frac\,, ''i'' denotes the interior product, ''dx''ω is the exterior derivative of ω with respect to the space variables only and \dot\, is the time-derivative of ω.== Proof of Theorem ==:Lemma. One has:::\frac \left (\int_a^b f(x)\; \mathrmx \right ) = f(b), \qquad \frac \left (\int_a^b f(x)\; \mathrmx \right )= -f(a).Proof. From proof of the fundamental theorem of calculus,:\begin \frac \left (\int_a^b f(x)\; \mathrmx \right ) &= \lim_ \frac \left(\int_a^ f(x)\,\mathrmx - \int_a^b f(x)\,\mathrmx \right ) \\ &= \lim_ \frac \int_b^ f(x)\,\mathrmx \\ &= \lim_ \frac \left(f(b) \Delta b + \mathcal\left(\Delta b^2\right) \right ) \\ &= f(b) \\ \frac \left (\int_a^b f(x)\; \mathrmx \right )&= \lim_ \frac \left(\int_^b f(x)\,\mathrmx - \int_a^b f(x)\,\mathrmx \right ) \\ &= \lim_ \frac \int_^a f(x)\,\mathrmx \\ &= \lim_ \frac \left(-f(a)\, \Delta a + \mathcal\left(\Delta a^2\right) \right )\\ &= -f(a).\endSuppose ''a'' and ''b'' are constant, and that ''f''(''x'') involves a parameter α which is constant in the integration but may vary to form different integrals. Assuming that ''f''(''x'', α) is a continuous function of ''x'' and α in the compact set , and that the partial derivative ''f''α(''x'', α) exists and is continuous, then if one defines::\varphi(\alpha) = \int_a^b f(x,\alpha)\;\mathrmx.:\varphi may be differentiated with respect to α by differentiating under the integral sign; i.e.,:\frac=\int_a^b\frac\,f(x,\alpha)\,\mathrmx.\,By the Heine–Cantor theorem it is uniformly continuous in that set. In other words for any ε > 0 there exists Δα such that for all values of ''x'' in (''b'' )::|f(x,\alpha+\Delta \alpha)-f(x,\alpha)|On the other hand::\begin\Delta\varphi &=\varphi(\alpha+\Delta \alpha)-\varphi(\alpha) \\&=\int_a^b f(x,\alpha+\Delta\alpha)\;\mathrmx - \int_a^b f(x,\alpha)\; \mathrmx \\&=\int_a^b \left (f(x,\alpha+\Delta\alpha)-f(x,\alpha) \right )\;\mathrmx \\&\leq \varepsilon (b-a)\endHence φ(α) is a continuous function.Similarly if \frac\,f(x,\alpha) exists and is continuous, then for all ε > 0 there exists Δα such that::\forall x \in (b ) \quad \left|\frac - \frac\right|Therefore,:\frac=\int_a^b\frac\;\mathrmx = \int_a^b \frac\,\mathrmx + Rwhere:|R| Now, ε → 0 as Δα → 0, therefore,:\lim_ \rarr 0}\frac= \frac = \int_a^b \frac\,f(x,\alpha)\,\mathrmx.\,This is the formula we set out to prove.Now, suppose:\int_a^b f(x,\alpha)\;\mathrmx=\varphi(\alpha),where ''a'' and ''b'' are functions of α which take increments Δ''a'' and Δ''b'', respectively, when α is increased by Δα. Then,:\begin\Delta\varphi &=\varphi(\alpha+\Delta\alpha)-\varphi(\alpha) \\&=\int_^f(x,\alpha+\Delta\alpha)\;\mathrmx\,-\int_a^b f(x,\alpha)\;\mathrmx\, \\&=\int_^af(x,\alpha+\Delta\alpha)\;\mathrmx+\int_a^bf(x,\alpha+\Delta\alpha)\;\mathrmx+\int_b^f(x,\alpha+\Delta\alpha)\;\mathrmx -\int_a^b f(x,\alpha)\;\mathrmx \\&=-\int_a^\,f(x,\alpha+\Delta\alpha)\;\mathrmx+\int_a^b()\;\mathrmx+\int_b^\,f(x,\alpha+\Delta\alpha)\;\mathrmx.\endA form of the mean value theorem, \int_a^bf(x)\;\mathrmx=(b-a)f(\xi), where ''a'' :\Delta\varphi=-\Delta a\,f(\xi_1,\alpha+\Delta\alpha)+\int_a^b()\;\mathrmx+\Delta b\,f(\xi_2,\alpha+\Delta\alpha).Dividing by Δα, letting Δα → 0, noticing ξ1 → ''a'' and ξ2 → ''b'' and using the above derivation for:\frac = \int_a^b\frac\,f(x,\alpha)\,\mathrmxyields:\frac = \int_a^b\frac\,f(x,\alpha)\,\mathrmx+f(b,\alpha)\frac-f(a,\alpha)\frac. This is the general form of the Leibniz integral rule.

Differentiation under the integral sign is a useful operation in calculus. Formally it can be stated as follows:
:Theorem. Let ''f''(''x'', ''t'') be a function such that both ''f''(''x'', ''t'') and its partial derivative ''fx''(''x'', ''t'') are continuous in ''t'' and ''x'' in some region of the (''x'', ''t'')-plane, including ''a''(''x'') ≤ ''t'' ≤ ''b''(''x''), ''x''0 ≤ ''x'' ≤ ''x''1. Also suppose that the functions ''a''(''x'') and ''b''(''x'') are both continuous and both have continuous derivatives for ''x''0 ≤ ''x'' ≤ ''x''1. Then for ''x''0 ≤ ''x'' ≤ ''x''1:
::\fracx} \left (\int_^f(x,t)\,\mathrmt \right) = f(x,b(x))\cdot b'(x) - f(x,a(x))\cdot a'(x) + \int_^ \fracf(x,t)\; \mathrmt.
This formula is the general form of the Leibniz integral rule and can be derived using the fundamental theorem of calculus. The (first) fundamental theorem of calculus is just a particular case of the above formula, for ''a''(''x'') = ''a'', a constant, ''b''(''x'') = ''x'' and ''f''(''x'', ''t'') = ''f''(''t'').
If both upper and lower limits are taken as constants, then the formula takes the shape of an operator equation:
:''ItDx'' = ''DxIt'',
where ''Dx'' is the partial derivative with respect to ''x'' and ''It'' is the integral operator with respect to ''t'' over a fixed interval. That is, it is related to the symmetry of second derivatives, but involving integrals as well as derivatives. This case is also known as the Leibniz integral rule.
The following three basic theorems on the interchange of limits are essentially equivalent:
* the interchange of a derivative and an integral (differentiation under the integral sign; i.e., Leibniz integral rule)
* the change of order of partial derivatives
* the change of order of integration (integration under the integral sign; i.e., Fubini's theorem)
== Higher dimensions ==
The Leibniz integral rule can be extended to multidimensional integrals. In two and three dimensions, this rule is better known from the field of fluid dynamics as the Reynolds transport theorem:
:\fract} \int_ F(\vec, t) \,\mathrmV = \int_ \frac \,F(\vec, t)\,\mathrmV + \int_ \,F(\vec, t)\, \vec_b \cdot \mathrm\mathbf
where F(\vec, t)\, is a scalar function, ''D''(''t'') and ∂''D''(''t'') denote a time-varying connected region of R3 and its boundary, respectively, \vec_b\, is the Eulerian velocity of the boundary (see Lagrangian and Eulerian coordinates) and dΣ = n dS is the unit normal component of the surface element.
The general statement of the Leibniz integral rule requires concepts from differential geometry, specifically differential forms, exterior derivatives, wedge products and interior products. With those tools, the Leibniz integral rule in ''p''-dimensions is:〔,〕
:\fract}\int_\omega=\int_ i__x\omega)+\int_ i_\dot,\,
where Ω(''t'') is a time-varying domain of integration, ω is a ''p''-form, \vec\, is the vector field of the velocity, \vec=\frac\,, ''i'' denotes the interior product, ''dx''ω is the exterior derivative of ω with respect to the space variables only and \dot\, is the time-derivative of ω.
== Proof of Theorem ==
:Lemma. One has:
::\frac \left (\int_a^b f(x)\; \mathrmx \right ) = f(b), \qquad \frac \left (\int_a^b f(x)\; \mathrmx \right )= -f(a).
Proof. From proof of the fundamental theorem of calculus,
:\begin
\frac \left (\int_a^b f(x)\; \mathrmx \right ) &= \lim_ \frac \left(\int_a^ f(x)\,\mathrmx - \int_a^b f(x)\,\mathrmx \right ) \\
&= \lim_ \frac \int_b^ f(x)\,\mathrmx \\
&= \lim_ \frac \left(f(b) \Delta b + \mathcal\left(\Delta b^2\right) \right ) \\
&= f(b) \\
\frac \left (\int_a^b f(x)\; \mathrmx \right )&= \lim_ \frac \left(\int_^b f(x)\,\mathrmx - \int_a^b f(x)\,\mathrmx \right ) \\
&= \lim_ \frac \int_^a f(x)\,\mathrmx \\
&= \lim_ \frac \left(-f(a)\, \Delta a + \mathcal\left(\Delta a^2\right) \right )\\
&= -f(a).
\end
Suppose ''a'' and ''b'' are constant, and that ''f''(''x'') involves a parameter α which is constant in the integration but may vary to form different integrals. Assuming that ''f''(''x'', α) is a continuous function of ''x'' and α in the compact set , and that the partial derivative ''f''α(''x'', α) exists and is continuous, then if one defines:
:\varphi(\alpha) = \int_a^b f(x,\alpha)\;\mathrmx.
:\varphi may be differentiated with respect to α by differentiating under the integral sign; i.e.,
:\frac=\int_a^b\frac\,f(x,\alpha)\,\mathrmx.\,
By the Heine–Cantor theorem it is uniformly continuous in that set. In other words for any ε > 0 there exists Δα such that for all values of ''x'' in (''b'' ):
:|f(x,\alpha+\Delta \alpha)-f(x,\alpha)|<\varepsilon.
On the other hand:
:\begin
\Delta\varphi &=\varphi(\alpha+\Delta \alpha)-\varphi(\alpha) \\
&=\int_a^b f(x,\alpha+\Delta\alpha)\;\mathrmx - \int_a^b f(x,\alpha)\; \mathrmx \\
&=\int_a^b \left (f(x,\alpha+\Delta\alpha)-f(x,\alpha) \right )\;\mathrmx \\
&\leq \varepsilon (b-a)
\end
Hence φ(α) is a continuous function.
Similarly if \frac\,f(x,\alpha) exists and is continuous, then for all ε > 0 there exists Δα such that:
:\forall x \in (b ) \quad \left|\frac - \frac\right|<\varepsilon.
Therefore,
:\frac=\int_a^b\frac\;\mathrmx = \int_a^b \frac\,\mathrmx + R
where
:|R| < \int_a^b \varepsilon\; \mathrmx = \varepsilon(b-a).
Now, ε → 0 as Δα → 0, therefore,
:\lim_ \rarr 0}\frac= \frac = \int_a^b \frac\,f(x,\alpha)\,\mathrmx.\,
This is the formula we set out to prove.
Now, suppose
:\int_a^b f(x,\alpha)\;\mathrmx=\varphi(\alpha),
where ''a'' and ''b'' are functions of α which take increments Δ''a'' and Δ''b'', respectively, when α is increased by Δα. Then,
:\begin
\Delta\varphi &=\varphi(\alpha+\Delta\alpha)-\varphi(\alpha) \\
&=\int_^f(x,\alpha+\Delta\alpha)\;\mathrmx\,-\int_a^b f(x,\alpha)\;\mathrmx\, \\
&=\int_^af(x,\alpha+\Delta\alpha)\;\mathrmx+\int_a^bf(x,\alpha+\Delta\alpha)\;\mathrmx+\int_b^f(x,\alpha+\Delta\alpha)\;\mathrmx -\int_a^b f(x,\alpha)\;\mathrmx \\
&=-\int_a^\,f(x,\alpha+\Delta\alpha)\;\mathrmx+\int_a^b()\;\mathrmx+\int_b^\,f(x,\alpha+\Delta\alpha)\;\mathrmx.
\end
A form of the mean value theorem, \int_a^bf(x)\;\mathrmx=(b-a)f(\xi), where ''a'' < ξ < ''b'', can be applied to the first and last integrals of the formula for Δφ above, resulting in
:\Delta\varphi=-\Delta a\,f(\xi_1,\alpha+\Delta\alpha)+\int_a^b()\;\mathrmx+\Delta b\,f(\xi_2,\alpha+\Delta\alpha).
Dividing by Δα, letting Δα → 0, noticing ξ1 → ''a'' and ξ2 → ''b'' and using the above derivation for
:\frac = \int_a^b\frac\,f(x,\alpha)\,\mathrmx
yields
:\frac = \int_a^b\frac\,f(x,\alpha)\,\mathrmx+f(b,\alpha)\frac-f(a,\alpha)\frac.
This is the general form of the Leibniz integral rule.

抄文引用元・出典: フリー百科事典『 ウィキペディア(Wikipedia)
ウィキペディアでDifferentiation under the integral sign is a useful operation in calculus. Formally it can be stated as follows::Theorem. Let ''f''(''x'', ''t'') be a function such that both ''f''(''x'', ''t'') and its partial derivative ''fx''(''x'', ''t'') are continuous in ''t'' and ''x'' in some region of the (''x'', ''t'')-plane, including ''a''(''x'') ≤ ''t'' ≤ ''b''(''x''), ''x''0 ≤ ''x'' ≤ ''x''1. Also suppose that the functions ''a''(''x'') and ''b''(''x'') are both continuous and both have continuous derivatives for ''x''0 ≤ ''x'' ≤ ''x''1. Then for ''x''0 ≤ ''x'' ≤ ''x''1:::\fracx} \left (\int_^f(x,t)\,\mathrmt \right) = f(x,b(x))\cdot b'(x) - f(x,a(x))\cdot a'(x) + \int_^ \fracf(x,t)\; \mathrmt.This formula is the general form of the Leibniz integral rule and can be derived using the fundamental theorem of calculus. The (first) fundamental theorem of calculus is just a particular case of the above formula, for ''a''(''x'') = ''a'', a constant, ''b''(''x'') = ''x'' and ''f''(''x'', ''t'') = ''f''(''t'').If both upper and lower limits are taken as constants, then the formula takes the shape of an operator equation::''ItDx'' = ''DxIt'',where ''Dx'' is the partial derivative with respect to ''x'' and ''It'' is the integral operator with respect to ''t'' over a fixed interval. That is, it is related to the symmetry of second derivatives, but involving integrals as well as derivatives. This case is also known as the Leibniz integral rule.The following three basic theorems on the interchange of limits are essentially equivalent:* the interchange of a derivative and an integral (differentiation under the integral sign; i.e., Leibniz integral rule)* the change of order of partial derivatives* the change of order of integration (integration under the integral sign; i.e., Fubini's theorem)== Higher dimensions == The Leibniz integral rule can be extended to multidimensional integrals. In two and three dimensions, this rule is better known from the field of fluid dynamics as the Reynolds transport theorem::\fract} \int_ F(\vec, t) \,\mathrmV = \int_ \frac \,F(\vec, t)\,\mathrmV + \int_ \,F(\vec, t)\, \vec_b \cdot \mathrm\mathbfwhere F(\vec, t)\, is a scalar function, ''D''(''t'') and ∂''D''(''t'') denote a time-varying connected region of R3 and its boundary, respectively, \vec_b\, is the Eulerian velocity of the boundary (see Lagrangian and Eulerian coordinates) and dΣ = n dS is the unit normal component of the surface element.The general statement of the Leibniz integral rule requires concepts from differential geometry, specifically differential forms, exterior derivatives, wedge products and interior products. With those tools, the Leibniz integral rule in ''p''-dimensions is:,:\fract}\int_\omega=\int_ i__x\omega)+\int_ i_\dot,\,where Ω(''t'') is a time-varying domain of integration, ω is a ''p''-form, \vec\, is the vector field of the velocity, \vec=\frac\,, ''i'' denotes the interior product, ''dx''ω is the exterior derivative of ω with respect to the space variables only and \dot\, is the time-derivative of ω.== Proof of Theorem ==:Lemma. One has:::\frac \left (\int_a^b f(x)\; \mathrmx \right ) = f(b), \qquad \frac \left (\int_a^b f(x)\; \mathrmx \right )= -f(a).Proof. From proof of the fundamental theorem of calculus,:\begin \frac \left (\int_a^b f(x)\; \mathrmx \right ) &= \lim_ \frac \left(\int_a^ f(x)\,\mathrmx - \int_a^b f(x)\,\mathrmx \right ) \\ &= \lim_ \frac \int_b^ f(x)\,\mathrmx \\ &= \lim_ \frac \left(f(b) \Delta b + \mathcal\left(\Delta b^2\right) \right ) \\ &= f(b) \\ \frac \left (\int_a^b f(x)\; \mathrmx \right )&= \lim_ \frac \left(\int_^b f(x)\,\mathrmx - \int_a^b f(x)\,\mathrmx \right ) \\ &= \lim_ \frac \int_^a f(x)\,\mathrmx \\ &= \lim_ \frac \left(-f(a)\, \Delta a + \mathcal\left(\Delta a^2\right) \right )\\ &= -f(a).\endSuppose ''a'' and ''b'' are constant, and that ''f''(''x'') involves a parameter α which is constant in the integration but may vary to form different integrals. Assuming that ''f''(''x'', α) is a continuous function of ''x'' and α in the compact set , and that the partial derivative ''f''α(''x'', α) exists and is continuous, then if one defines::\varphi(\alpha) = \int_a^b f(x,\alpha)\;\mathrmx.:\varphi may be differentiated with respect to α by differentiating under the integral sign; i.e.,:\frac=\int_a^b\frac\,f(x,\alpha)\,\mathrmx.\,By the Heine–Cantor theorem it is uniformly continuous in that set. In other words for any ε > 0 there exists Δα such that for all values of ''x'' in (''b'' )::|f(x,\alpha+\Delta \alpha)-f(x,\alpha)|On the other hand::\begin\Delta\varphi &=\varphi(\alpha+\Delta \alpha)-\varphi(\alpha) \\&=\int_a^b f(x,\alpha+\Delta\alpha)\;\mathrmx - \int_a^b f(x,\alpha)\; \mathrmx \\&=\int_a^b \left (f(x,\alpha+\Delta\alpha)-f(x,\alpha) \right )\;\mathrmx \\&\leq \varepsilon (b-a)\endHence φ(α) is a continuous function.Similarly if \frac\,f(x,\alpha) exists and is continuous, then for all ε > 0 there exists Δα such that::\forall x \in (b ) \quad \left|\frac - \frac\right|Therefore,:\frac=\int_a^b\frac\;\mathrmx = \int_a^b \frac\,\mathrmx + Rwhere:|R| Now, ε → 0 as Δα → 0, therefore,:\lim_ \rarr 0}\frac= \frac = \int_a^b \frac\,f(x,\alpha)\,\mathrmx.\,This is the formula we set out to prove.Now, suppose:\int_a^b f(x,\alpha)\;\mathrmx=\varphi(\alpha),where ''a'' and ''b'' are functions of α which take increments Δ''a'' and Δ''b'', respectively, when α is increased by Δα. Then,:\begin\Delta\varphi &=\varphi(\alpha+\Delta\alpha)-\varphi(\alpha) \\&=\int_^f(x,\alpha+\Delta\alpha)\;\mathrmx\,-\int_a^b f(x,\alpha)\;\mathrmx\, \\&=\int_^af(x,\alpha+\Delta\alpha)\;\mathrmx+\int_a^bf(x,\alpha+\Delta\alpha)\;\mathrmx+\int_b^f(x,\alpha+\Delta\alpha)\;\mathrmx -\int_a^b f(x,\alpha)\;\mathrmx \\&=-\int_a^\,f(x,\alpha+\Delta\alpha)\;\mathrmx+\int_a^b()\;\mathrmx+\int_b^\,f(x,\alpha+\Delta\alpha)\;\mathrmx.\endA form of the mean value theorem, \int_a^bf(x)\;\mathrmx=(b-a)f(\xi), where ''a'' :\Delta\varphi=-\Delta a\,f(\xi_1,\alpha+\Delta\alpha)+\int_a^b()\;\mathrmx+\Delta b\,f(\xi_2,\alpha+\Delta\alpha).Dividing by Δα, letting Δα → 0, noticing ξ1 → ''a'' and ξ2 → ''b'' and using the above derivation for:\frac = \int_a^b\frac\,f(x,\alpha)\,\mathrmxyields:\frac = \int_a^b\frac\,f(x,\alpha)\,\mathrmx+f(b,\alpha)\frac-f(a,\alpha)\frac. This is the general form of the Leibniz integral rule.」の詳細全文を読む



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