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
location=New York

 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.


Basic form

Let us first make the assignment

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


: {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.


::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

Wikimedia Foundation. 2010.

Look at other dictionaries:

  • Leibniz (disambiguation) — Gottfried Leibniz (1646 – 1716) was a German philosopher and mathematician.In mathematics, several results and concepts are attributed to Leibniz:* Leibniz algebra, an algebraic structure * Leibniz formula for pi, a method for calculating pi; *… …   Wikipedia

  • Integral — This article is about the concept of integrals in calculus. For the set of numbers, see integer. For other uses, see Integral (disambiguation). A definite integral of a function can be represented as the signed area of the region bounded by its… …   Wikipedia

  • Leibniz's notation — In calculus, Leibniz s notation, named in honor of the 17th century German philosopher and mathematician Gottfried Wilhelm Leibniz, was originally the use of expressions such as d x and d y and to represent infinitely small (or infinitesimal)… …   Wikipedia

  • Differentiation under the integral sign — Topics in Calculus Fundamental theorem Limits of functions Continuity Mean value theorem Differential calculus  Derivative Change of variables Implicit differentiation Taylor s theorem Related rates …   Wikipedia

  • Gottfried Leibniz — Infobox Philosopher region = Western Philosophy era = 18th century philosophy color = #B0C4DE |250px image caption = Gottfried Wilhelm Leibniz name = Gottfried Wilhelm Leibniz birth = 1 July (21 June Old Style) 1646, Leipzig, Electorate of Saxony …   Wikipedia

  • Chain rule — For other uses, see Chain rule (disambiguation). Topics in Calculus Fundamental theorem Limits of functions Continuity Mean value theorem Differential calculus  Derivative Change of variables Implicit differentiation …   Wikipedia

  • Dirichlet integral — In mathematics, there are several integrals known as the Dirichlet integral, after the German mathematician Peter Gustav Lejeune Dirichlet. One of those is This can be derived from attempts to evaluate a double improper integral two different… …   Wikipedia

  • Product rule — For Euler s chain rule relating partial derivatives of three independent variables, see Triple product rule. For the counting principle in combinatorics, see Rule of product. Topics in Calculus Fundamental theorem Limits of functions Continuity… …   Wikipedia

  • List of mathematics articles (L) — NOTOC L L (complexity) L BFGS L² cohomology L function L game L notation L system L theory L Analyse des Infiniment Petits pour l Intelligence des Lignes Courbes L Hôpital s rule L(R) La Géométrie Labeled graph Labelled enumeration theorem Lack… …   Wikipedia

  • Luke's variational principle — In fluid dynamics, Luke s variational principle is a Lagrangian variational description of the motion of surface waves on a fluid with a free surface, under the action of gravity. This principle is named after J.C. Luke, who published it in 1967 …   Wikipedia

Share the article and excerpts

Direct link
Do a right-click on the link above
and select “Copy Link”

We are using cookies for the best presentation of our site. Continuing to use this site, you agree with this.