Leibniz integral rule

In mathematics, Leibniz's rule for differentiation under the integral sign, named after Gottfried Leibniz, tells us that if we have an integral of the form

: $int\_\{y\_0\}^\{y\_1\}\; f(x,\; y)\; ,dy$

then for $x\; in\; (x\_0,\; x\_1)$ the derivative of this integral is thus expressible

: $\{dover\; dx\},\; int\_\{y\_0\}^\{y\_1\}\; f(x,\; y)\; ,dy\; =\; int\_\{y\_0\}^\{y\_1\}\; \{partial\; over\; partial\; x\}\; f(x,y),dy$

provided that $f$ and $partial\; f\; /\; partial\; x$ are both continuous over a region in the form

:$[x\_0,x\_1]\; imes\; [y\_0,y\_1]\; .,$

Limits that are variable

A more general result, applicable when the limits of integration "a" and "b" and the integrand "f" ( "x", α ) all are functions of the parameter α is:

  $frac\{d\}\{dalpha\}int\_a^b\; f(x,alpha)dx\; =\; int\_a^bfrac\{partial\}\{partial\; alpha\},f(x,alpha),dx+f(b,alpha)frac\{d\; b\}\{d\; alpha\}-f(a,alpha)frac\{d\; a\}\{d\; alpha\},$  

where the partial derivative of "f" indicates that inside the integral only the variation of "f" ( "x", α ) with α is considered in taking the derivative.

Three-dimensional, time-dependent case

A Leibniz integral rule for three dimensions is:cite book
author=Heinz Knoepfel
title=Magnetic fields: A comprehensive theoretical treatise for practical use
year= 2000
page =Eq. 1.4-11, p. 36
publisher=Wiley-IEEE
location=New York
isbn=0471322059
url=http://books.google.com/books?hl=en&lr=&id=HlsGlaxe5twC&oi=fnd&pg=PA1&dq=moving+media+%22Faraday%27s+law+%22&ots=ZKzsORjIwe&sig=QsTNgfeSv53Fbv-aJ-4pKdOvjNw#PPA36,M1]

 $frac\; \{d\}\{dt\}\; iint\_\{\; Sigma\; (t)\}\; mathbf\{F\}\; (mathbf\{r\},\; t)\; cdot\; d\; mathbf\{A\}$

  $=\; iint\_\{Sigma\; (t)\}left\; [\; frac\; \{partial\}\{partial\; t\}mathbf\{F\}\; (mathbf\{r\},\; t)\; +\; left(mathrm\{\; abla\}\; cdot\; mathbf\{F\}\; (mathbf\{r\},\; t)\; ight)\; mathbf\{v\}\; ight]\; cdot\; d\; mathbf\{A\}$::::::$-\; oint\_\{partial\; Sigma\; (t)\}\; left(\; mathbf\{v\; imes\; \}mathbf\{F\}\; (\; mathbf\{r\},\; t)\; ight)\; cdot\; d\; mathbf\{s\}\; ,$

where:::F ( r, t ) is a vector field at the spatial position r at time "t"::Σ is a moving surface in three-space bounded by the closed curve ∂Σ::"d" A is a vector element of the surface Σ::"d" s is a vector element of the curve ∂Σ::v is the velocity of movement of the region Σ::∇• is the vector divergence::× is the vector cross product:The double integrals are surface integrals over the surface Σ, and the line integral is over the bounding curve ∂Σ.

Proofs

Basic form

Let us first make the assignment

: $u(x)\; =\; int\_\{y\_0\}^\{y\_1\}\; f(x,\; y)\; ,dy.$

Then

: $\{dover\; dx\}\; u(x)\; =\; lim\_\{h\; ightarrow\; 0\}\; \{u(x+h)-u(x)\; over\; h\}.$

Substituting back

: $=\; lim\_\{h\; ightarrow\; 0\}\; \{int\_\{y\_0\}^\{y\_1\}f(x+h,y),dy-int\_\{y\_0\}^\{y\_1\}f(x,y),dy\; over\; h\}.$

Since integration is linear, we can write the two integrals as one:

: $=\; lim\_\{h\; ightarrow\; 0\}\; \{int\_\{y\_0\}^\{y\_1\}(f(x+h,y)-f(x,y)),dyover\; h\}.$

And we can take the constant inside, with the integrand

: $=\; lim\_\{h\; ightarrow\; 0\}\; int\_\{y\_0\}^\{y\_1\}\; \{f(x+h,y)-f(x,y)over\; h\},dy.$

And now, since the integrand is in the form of a difference quotient:

: $=\; int\_\{y\_0\}^\{y\_1\}\; \{partial\; over\; partial\; x\}\; f(x,y),dy$

which can be justified by uniform continuity, and therefore

: $\{dover\; dx\}\; u(x)\; =\; int\_\{y\_0\}^\{y\_1\}\; \{partial\; over\; partial\; x\}\; f(x,\; y)\; ,dy.$

Variable limits form

For a monovariant function $g$:

: $\{dover\; dx\},\; int\_\{f\_1(x)\}^\{f\_2(x)\}\; g(t)\; ,dt\; =\; g(f\_2(x))\; \{f\_2\text{'}(x)\}\; -\; g(f\_1(x))\; \{f\_1\text{'}(x)\}$

This follows from the chain rule.

General form with variable limits

Now, suppose $int\_a^b\; f(x,alpha)dx=phi(alpha),$, where "a" and "b" are functions of α that exhibit increments Δ"a" and Δ"b", respectively, when α is increased by Δα.

Then,

::$Deltaphi=phi(alpha+Deltaalpha)-phi(alpha)=int\_\{a+Delta\; a\}^\{b+Delta\; b\}f(x,alpha+Deltaalpha)dx,-int\_a^b\; f(x,alpha)dx,$

::$=int\_\{a+Delta\; a\}^af(x,alpha+Deltaalpha)dx+int\_a^bf(x,alpha+Deltaalpha)dx+int\_b^\{b+Delta\; b\}f(x,alpha+Deltaalpha)dx,-int\_a^b\; f(x,alpha)dx,$

::$=-int\_a^\{a+Delta\; a\},f(x,alpha+Deltaalpha)dx+int\_a^b\; [f(x,alpha+Deltaalpha)-f(x,alpha)]\; dx+int\_b^\{b+Delta\; b\},f(x,alpha+Deltaalpha)dx,$.

A form of the mean value theorem, $int\_a^bf(x)dx=(b-a)f(xi),$, where

::$Deltaphi=-Delta\; a,f(xi\_1,alpha+Deltaalpha)+int\_a^b\; [f(x,alpha+Deltaalpha)-f(x,alpha)]\; dx+Delta\; b,f(xi\_2,alpha+Deltaalpha),$.

Dividing by $Deltaalpha,$, and letting $Deltaalpha\; arr0,$, and noticing $xi\_1\; arr\; a,$ and $xi\_2\; arr\; b,$ and using the result ::$frac\{dphi\}\{dalpha\}\; =\; int\_a^b\; frac\{partial\}\; \{partial\; alpha\}\; ,f(x,alpha),dx$ yields the general form of the Leibniz integral rule below:

::$frac\{dphi\}\{dalpha\}\; =\; int\_a^bfrac\{partial\}\{partial\; alpha\},f(x,alpha),dx+f(b,alpha)frac\{partial\; b\}\{partial\; alpha\}-f(a,alpha)frac\{partial\; a\}\{partial\; alpha\}\; .$

Three-dimensional, time-dependent form

At time "t" the surface Σ in Figure 1 contains a set of points arranged about a centroid R ( "t" ) and function F ( r, "t") can be written as F ( R ( "t" ) + r − R(t), "t" ) = F ( R ( "t" ) + ρ, "t" ), with ρ independent of time. Variables are shifted to a new frame of reference attached to the moving surface, with origin at R ( "t" ). For a rigidly translating surface, the limits of integration are then independent of time, so:

:$frac\; \{d\}\{dt\}\; iint\_\{Sigma\; (t)\}\; d\; mathbf\{A\}\_\{mathbf\{rmathbf\{\; cdot\; F\; \}\; (\; mathbf\{r\}\; ,\; t\; )$ $=\; iint\_\{Sigma\; \}\; d\; mathbf\{A\}\_\{vec\{\; ho\; cdot\; frac\; \{d\}\{dt\}mathbf\{F\}(\; mathbf\{R\}(t)\; +\; vec\{\; ho\},\; t)$

where the limits of integration confining the integral to the region Σ no longer are time dependent so differentiation passes through the integration to act on the integrand only: ::$frac\; \{d\}\{dt\}mathbf\{F\}(\; mathbf\{R\}(t)\; +\; vec\; ho\; ,\; t)\; =\; frac\; \{partial\}\{partial\; t\}mathbf\{F\}(\; mathbf\{R\}(t)\; +\; vec\{\; ho\},\; t)\; +\; mathbf\{v\; cdot\; abla\; F\}(\; mathbf\{R\}(t)\; +\; vec\{\; ho\},\; t)$

:::::::$=\; frac\; \{partial\}\{partial\; t\}mathbf\{F\}(\; mathbf\{r\},\; t)\; +\; mathbf\{v\; cdot\; abla\; F\}(\; mathbf\{r\},\; t)\; ,$with the velocity of motion of the surface defined by::$mathbf\{v\}\; =\; frac\; \{d\}\{dt\}\; mathbf\{R\}\; (t)\; .$

This equation expresses the material derivative of the field, that is, the derivative with respect to a coordinate system attached to the moving surface. Having found the derivative, variables can be switched back to the original frame of reference. We notice that (see article on "curl" ):

::$mathbf\{\; abla\; imes\}\; left(\; mathbf\{v\; imes\; F\}\; ight)\; =\; left\; [\; left(\; mathbf\{\; abla\; cdot\; F\; \}\; ight)\; +\; mathbf\{F\; cdot\; abla\}\; ight]\; mathbf\{v\}-\; left\; [\; left(\; mathbf\{\; abla\; cdot\; v\; \}\; ight)\; +\; mathbf\{v\; cdot\; abla\}\; ight]\; mathbf\{F\}\; .$

and that Stokes theorem allows the surface integral of the "curl" over Σ to be made a line integral over ∂Σ:

 $frac\; \{d\}\{dt\}\; iint\_\{\; Sigma\; (t)\}\; mathbf\{F\}\; (mathbf\{r\},\; t)\; cdot\; d\; mathbf\{A\}$::::$=\; iint\_\{\; Sigma\; (t)\}\; left\; [\; frac\; \{partial\}\{partial\; t\}\; mathbf\{F\}\; (mathbf\{r\},\; t)\; +left(\; mathbf\{F\; cdot\; abla\}\; ight)mathbf\{v\}\; +\; left(mathbf\{\; abla\; cdot\; F\; \}\; ight)\; mathbf\{v\}\; -(\; abla\; cdot\; mathbf\{v\})mathbf\{F\}\; ight]\; cdot\; d\; mathbf\{A\}\; -\; oint\_\{partial\; Sigma\; (t)\; \}left(\; mathbf\{mathbf\{v\}\; imes\; F\; \}\; ight)mathbf\{cdot\}\; d\; mathbf\{s\}\; .$

The sign of the line integral is based on the right-hand rule for the choice of direction of line element "d"s. To establish this sign, for example, suppose the field F points in the positive "z"-direction, and the surface Σ is a portion of the "xy"-plane with perimeter ∂Σ. We adopt the normal to Σ to be in the positive "z"-direction. Positive traversal of ∂Σ is then counterclockwise (right-hand rule with thumb along "z"-axis). Then the integral on the left-hand side determines a "positive" flux of F through Σ. Suppose Σ translates in the positive "x"-direction at velocity v. An element of the boundary of Σ parallel to the "y"-axis, say "d"s, sweeps out an area v"t" × "d"s in time "t". If we integrate around the boundary ∂Σ in a counterclockwise sense, v"t" × "d"s points in the negative "z"-direction on the left side of ∂Σ (where "d"s points downward), and in the positive "z"-direction on the right side of ∂Σ (where "d"s points upward), which makes sense because Σ is moving to the right, adding area on the right and losing it on the left. On that basis, the flux of F is increasing on the right of ∂Σ and decreasing on the left. However, the dot-product v × F • "d"s = −F × v• "d"s = −F • v × "d"s. Consequently, the sign of the line integral is taken as negative.

If v is a constant,

 $frac\; \{d\}\{dt\}\; iint\_\{\; Sigma\; (t)\}\; mathbf\{F\}\; (mathbf\{r\},\; t)\; cdot\; d\; mathbf\{A\}$::::$=\; iint\_\{\; Sigma\; (t)\}\; left\; [\; frac\; \{partial\}\{partial\; t\}\; mathbf\{F\}\; (mathbf\{r\},\; t)\; +\; left(mathbf\{\; abla\; cdot\; F\; \}\; ight)\; mathbf\{v\}\; ight]\; cdot\; d\; mathbf\{A\}\; -\; oint\_\{partial\; Sigma\; (t)\}left(\; mathbf\{mathbf\{v\}\; imes\; F\; \}\; ight)mathbf\{cdot\}\; d\; mathbf\{s\}\; ,$

which is the quoted result. This proof does not consider the possibility of the surface deforming as it moves.

References and notes

ee also

*Chain rule
*Leibniz rule (generalized product rule)
*Differentiation under the integral sign