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# 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

: $int_\left\{y_0\right\}^\left\{y_1\right\} f\left(x, y\right) ,dy$

then for $x in \left(x_0, x_1\right)$ the derivative of this integral is thus expressible

: $\left\{dover dx\right\}, int_\left\{y_0\right\}^\left\{y_1\right\} f\left(x, y\right) ,dy = int_\left\{y_0\right\}^\left\{y_1\right\} \left\{partial over partial x\right\} f\left(x,y\right),dy$

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

:$\left[x_0,x_1\right] imes \left[y_0,y_1\right] .,$

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:

&emsp; $frac\left\{d\right\}\left\{dalpha\right\}int_a^b f\left(x,alpha\right)dx = int_a^bfrac\left\{partial\right\}\left\{partial alpha\right\},f\left(x,alpha\right),dx+f\left(b,alpha\right)frac\left\{d b\right\}\left\{d alpha\right\}-f\left(a,alpha\right)frac\left\{d a\right\}\left\{d alpha\right\},$ &emsp;

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
]

&emsp;$frac \left\{d\right\}\left\{dt\right\} iint_\left\{ Sigma \left(t\right)\right\} mathbf\left\{F\right\} \left(mathbf\left\{r\right\}, t\right) cdot d mathbf\left\{A\right\}$

&ensp;&ensp;$= iint_\left\{Sigma \left(t\right)\right\}left \left[ frac \left\{partial\right\}\left\{partial t\right\}mathbf\left\{F\right\} \left(mathbf\left\{r\right\}, t\right) + left\left(mathrm\left\{ abla\right\} cdot mathbf\left\{F\right\} \left(mathbf\left\{r\right\}, t\right) ight\right) mathbf\left\{v\right\} ight\right] cdot d mathbf\left\{A\right\}$::::::$- oint_\left\{partial Sigma \left(t\right)\right\} left\left( mathbf\left\{v imes \right\}mathbf\left\{F\right\} \left( mathbf\left\{r\right\}, t\right) ight\right) cdot d mathbf\left\{s\right\} ,$

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

: $u\left(x\right) = int_\left\{y_0\right\}^\left\{y_1\right\} f\left(x, y\right) ,dy.$

Then

: $\left\{dover dx\right\} u\left(x\right) = lim_\left\{h ightarrow 0\right\} \left\{u\left(x+h\right)-u\left(x\right) over h\right\}.$

Substituting back

: $= lim_\left\{h ightarrow 0\right\} \left\{int_\left\{y_0\right\}^\left\{y_1\right\}f\left(x+h,y\right),dy-int_\left\{y_0\right\}^\left\{y_1\right\}f\left(x,y\right),dy over h\right\}.$

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

: $= lim_\left\{h ightarrow 0\right\} \left\{int_\left\{y_0\right\}^\left\{y_1\right\}\left(f\left(x+h,y\right)-f\left(x,y\right)\right),dyover h\right\}.$

And we can take the constant inside, with the integrand

: $= lim_\left\{h ightarrow 0\right\} int_\left\{y_0\right\}^\left\{y_1\right\} \left\{f\left(x+h,y\right)-f\left(x,y\right)over h\right\},dy.$

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

: $= int_\left\{y_0\right\}^\left\{y_1\right\} \left\{partial over partial x\right\} f\left(x,y\right),dy$

which can be justified by uniform continuity, and therefore

: $\left\{dover dx\right\} u\left(x\right) = int_\left\{y_0\right\}^\left\{y_1\right\} \left\{partial over partial x\right\} f\left(x, y\right) ,dy.$

Variable limits form

For a monovariant function $g$:

: $\left\{dover dx\right\}, int_\left\{f_1\left(x\right)\right\}^\left\{f_2\left(x\right)\right\} g\left(t\right) ,dt = g\left(f_2\left(x\right)\right) \left\{f_2\text{'}\left(x\right)\right\} - g\left(f_1\left(x\right)\right) \left\{f_1\text{'}\left(x\right)\right\}$

This follows from the chain rule.

General form with variable limits

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

Then,

::$Deltaphi=phi\left(alpha+Deltaalpha\right)-phi\left(alpha\right)=int_\left\{a+Delta a\right\}^\left\{b+Delta b\right\}f\left(x,alpha+Deltaalpha\right)dx,-int_a^b f\left(x,alpha\right)dx,$

::$=int_\left\{a+Delta a\right\}^af\left(x,alpha+Deltaalpha\right)dx+int_a^bf\left(x,alpha+Deltaalpha\right)dx+int_b^\left\{b+Delta b\right\}f\left(x,alpha+Deltaalpha\right)dx,-int_a^b f\left(x,alpha\right)dx,$

::$=-int_a^\left\{a+Delta a\right\},f\left(x,alpha+Deltaalpha\right)dx+int_a^b \left[f\left(x,alpha+Deltaalpha\right)-f\left(x,alpha\right)\right] dx+int_b^\left\{b+Delta b\right\},f\left(x,alpha+Deltaalpha\right)dx,$.

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

::$Deltaphi=-Delta a,f\left(xi_1,alpha+Deltaalpha\right)+int_a^b \left[f\left(x,alpha+Deltaalpha\right)-f\left(x,alpha\right)\right] dx+Delta b,f\left(xi_2,alpha+Deltaalpha\right),$.

Dividing by $Deltaalpha,$, and letting $Deltaalpha arr0,$, and noticing $xi_1 arr a,$ and $xi_2 arr b,$ and using the result ::$frac\left\{dphi\right\}\left\{dalpha\right\} = int_a^b frac\left\{partial\right\} \left\{partial alpha\right\} ,f\left(x,alpha\right),dx$ yields the general form of the Leibniz integral rule below:

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

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:

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

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: ::$frac \left\{d\right\}\left\{dt\right\}mathbf\left\{F\right\}\left( mathbf\left\{R\right\}\left(t\right) + vec ho , t\right) = frac \left\{partial\right\}\left\{partial t\right\}mathbf\left\{F\right\}\left( mathbf\left\{R\right\}\left(t\right) + vec\left\{ ho\right\}, t\right) + mathbf\left\{v cdot abla F\right\}\left( mathbf\left\{R\right\}\left(t\right) + vec\left\{ ho\right\}, t\right)$

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

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" ):

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

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

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

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,

&emsp;$frac \left\{d\right\}\left\{dt\right\} iint_\left\{ Sigma \left(t\right)\right\} mathbf\left\{F\right\} \left(mathbf\left\{r\right\}, t\right) cdot d mathbf\left\{A\right\}$::::$= iint_\left\{ Sigma \left(t\right)\right\} left \left[ frac \left\{partial\right\}\left\{partial t\right\} mathbf\left\{F\right\} \left(mathbf\left\{r\right\}, t\right) + left\left(mathbf\left\{ abla cdot F \right\} ight\right) mathbf\left\{v\right\} ight\right] cdot d mathbf\left\{A\right\} - oint_\left\{partial Sigma \left(t\right)\right\}left\left( mathbf\left\{mathbf\left\{v\right\} imes F \right\} ight\right)mathbf\left\{cdot\right\} d mathbf\left\{s\right\} ,$

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

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