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'(x)} - g(f_1(x)) {f_1'(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 a, may be applied to the first and last integrals of the formula for Deltaphi, above, resulting in

::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


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