Leibniz integral rule

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In calculus, Leibniz's rule for differentiation under the integral sign, named after Gottfried Leibniz, states that for an integral of the form

where , the derivative of this integral is expressible as

where the partial derivative indicates that inside the integral, only the variation of f(x, t) with t is considered in taking the derivative.[1] Notice that if and are constants rather than functions of , we have a special case of Leibniz's rule:

Thus under certain conditions, one may interchange the integral and partial differential operators. This important result is particularly useful in the differentiation of integral transforms. An example of such is the moment generating function in probability theory, a variation of the Laplace transform, which can be differentiated to generate the moments of a random variable. Whether Leibniz's integral rule applies is essentially a question about the interchange of limits.

General form: Differentiation under the integral sign[edit]

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) ≤ tb(x), x0xx1. Also suppose that the functions a(x) and b(x) are both continuous and both have continuous derivatives for x0xx1. Then, for x0xx1,

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 the particular case of the above formula where 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).

Three-dimensional, time-dependent case[edit]

Figure 1: A vector field F(r, t) defined throughout space, and a surface Σ bounded by curve ∂Σ moving with velocity v over which the field is integrated.

A Leibniz integral rule for two dimensions is[2]

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,
dA is a vector element of the surface Σ,
ds 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.

Higher dimensions[edit]

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:

where is a scalar function, D(t) and ∂D(t) denote a time-varying connected region of R3 and its boundary, respectively, 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[2]

where Ω(t) is a time-varying domain of integration, ω is a p-form, is the vector field of the velocity, i denotes the interior product, dxω is the exterior derivative of ω with respect to the space variables only and is the time derivative of ω.

Measure theory statement[edit]

Let be an open subset of , and be a measure space. Suppose satisfies the following conditions:

  1. is a Lebesgue-integrable function of for each .
  2. For almost all , the derivative exists for all .
  3. There is an integrable function such that for all and almost every .

Then, for all ,

Proofs[edit]

Proof of basic form[edit]

Let

By the definition of the derivative,

Substitute equation (1) into equation (2). The difference of two integrals equals the integral of the difference, and 1/h is a constant, so

Provided that the limit can be passed through the integral sign, we obtain

We claim that the passage of the limit under the integral sign is valid by the bounded convergence theorem (a corollary of the dominated convergence theorem). For each δ > 0, consider the difference quotient

For t fixed, the mean value theorem implies there exists z in the interval [x, x + δ] such that

Continuity of fx(x, t) and compactness of the domain together imply that fx(x, t) is bounded. The above application of the mean value theorem therefore gives a uniform (independent of δ) bound on . The difference quotients converge pointwise to the partial derivative fx by the assumption that the partial derivative exists.

The above argument shows that for every sequence {δn} → 0, the sequence is uniformly bounded and converges pointwise to fx. The bounded convergence theorem states that if a sequence of functions on a set of finite measure is uniformly bounded and converges pointwise, then passage of the limit under the integral is valid. In particular, the limit and integral may be exchanged for every sequence {δn} → 0. Therefore the limit as δ → 0 may be passed through the integral sign.

For a simpler proof using Fubini's theorem, see the references.

Variable limits form[edit]

For a monovariant function g:

This follows from the chain rule.

General form with variable limits[edit]

Set

where a and b are functions of α that exhibit increments Δa and Δb, respectively, when α is increased by Δα. Then,

A form of the mean value theorem, , where a < ξ < b, may be applied to the first and last integrals of the formula for Δφ above, resulting in

Divide by Δα and let Δα → 0. Notice ξ1a and ξ2b. We may pass the limit through the integral sign:

again by the bounded convergence theorem. This yields the general form of the Leibniz integral rule,

Three-dimensional, time-dependent form[edit]

At time t the surface Σ in Figure 1 contains a set of points arranged about a centroid . The function can be written as

with independent of time. Variables are shifted to a new frame of reference attached to the moving surface, with origin at . For a rigidly translating surface, the limits of integration are then independent of time, so:

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:

with the velocity of motion of the surface defined by

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)

and that Stokes theorem equates the surface integral of the curl over Σ with a line integral over ∂Σ:

The sign of the line integral is based on the right-hand rule for the choice of direction of line element ds. 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 ds, sweeps out an area vt × ds in time t. If we integrate around the boundary ∂Σ in a counterclockwise sense, vt × ds points in the negative z-direction on the left side of ∂Σ (where ds points downward), and in the positive z-direction on the right side of ∂Σ (where ds 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 • ds = −F × vds = −F • v × ds. Consequently, the sign of the line integral is taken as negative.

If v is a constant,

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

Alternative derivation[edit]

Lemma. One has:

Proof. From proof of the fundamental theorem of calculus,

and

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. Assume that f(x, α) is a continuous function of x and α in the compact set {(x, α) : α0 ≤ α ≤ α1 and axb}, and that the partial derivative fα(x, α) exists and is continuous. If one defines:

then may be differentiated with respect to α by differentiating under the integral sign, i.e.,

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 [a, b],

On the other hand,

Hence φ(α) is a continuous function.

Similarly if exists and is continuous, then for all ε > 0 there exists Δα such that:

Therefore,

where

Now, ε → 0 as Δα → 0, so

This is the formula we set out to prove.

Now, suppose

where a and b are functions of α which take increments Δa and Δb, respectively, when α is increased by Δα. Then,

A form of the mean value theorem, where a < ξ < b, can be applied to the first and last integrals of the formula for Δφ above, resulting in

Dividing by Δα, letting Δα → 0, noticing ξ1a and ξ2b and using the above derivation for

yields

This is the general form of the Leibniz integral rule.

Examples[edit]

General examples[edit]

Example 1[edit]

Consider the function

The function under the integral sign is not continuous at the point (x, α) = (0, 0), and the function φ(α) has a discontinuity at α = 0 because φ(α) approaches ±π/2 as α → 0±.

If we differentiate φ(α) with respect to α under the integral sign, we get

which is, of course, true for all values of α except α = 0. This may be integrated (with respect to α) to find

Example 2[edit]

An example with variable limits:

Examples for evaluating a definite integral[edit]

Example 3[edit]

The principle of differentiating under the integral sign may sometimes be used to evaluate a definite integral. Consider:

Now,

As x varies from 0 to π, we have

Hence,

Therefore,

Integrating both sides with respect to α, we get:

C1 = 0 follows from evaluating φ(0):

To determine C2 in the same manner, we should need to substitute in a value of α greater than 1 in φ(α). This is somewhat inconvenient. Instead, we substitute α = 1/β, where |β| < 1. Then,

Therefore, C2 = 0.

The definition of φ(α) is now complete:

The foregoing discussion, of course, does not apply when α = ±1, since the conditions for differentiability are not met.

Example 4[edit]

First we calculate:

The limits of integration being independent of a, we have:

On the other hand:

Equating these two relations then yields

In a similar fashion, pursuing yields

Adding the two results then produces

which computes as desired.

This derivation may be generalized. Note that if we define

it can easily be shown that

Given I1, this integral reduction formula can be used to compute all of the values of In for n > 1.

Example 5[edit]

Here, we consider the integral

Differentiating under the integral with respect to α, we have

Therefore:

However, by definition, I(π/2) = 0, hence C = π2/8 and

Example 6[edit]

Here, we consider the integral

We introduce a new variable φ and rewrite the integral as

When φ = 1 this equals the original integral. However, this more general integral may be differentiated with respect to φ:

The interval of integration [0, 2π] is the union of the intervals [0, π] and [π, 2π], and therefore the integral is the sum of an integral over [0, π] and an integral over [π, 2π]. On these subintervals, the substitution u = φ cos θ is a diffeomorphism (except at the endpoints) and therefore does not change the value of the integral. This substitution sends 0 to φ, π to −φ, and 2π to φ. Therefore it sends both [0, π] and [π, 2π] to [−φ, φ], but with opposite orientations. That is,

This implies that f(φ) is constant. The constant may be determined by evaluating f at φ = 0:

Therefore the original integral also equals 2π.

Other problems to solve[edit]

There are innumerable other integrals that can be solved using the technique of differentiation under the integral sign. For example, in each of the following cases, the original integral may be replaced by a similar integral having a new parameter α:

The first integral, the Dirichlet integral, is absolutely convergent for positive α but only conditionally convergent when α is 0. Therefore, differentiation under the integral sign is easy to justify when α > 0, but proving that the resulting formula remains valid when α is 0 requires some careful work.

Applications to series[edit]

The measure-theoretic version of differentiation under the integral sign also applies to summation (finite or infinite) by interpreting summation as counting measure. An example of an application is the fact that power series are differentiable in their radius of convergence.

In popular culture[edit]

Differentiation under the integral sign is mentioned in the late physicist Richard Feynman's best-selling memoir Surely You're Joking, Mr. Feynman! in the chapter "A Different Box of Tools". He describes learning it, while in high school, from an old text, Advanced Calculus (1926), by Frederick S. Woods (who was a professor of mathematics in the Massachusetts Institute of Technology). The technique was not often taught when Feynman later received his formal education in calculus, but using it, Feynman was able to use the technique to solve otherwise difficult integration problems upon his arrival at graduate school at Princeton University:

One thing I never did learn was contour integration. I had learned to do integrals by various methods shown in a book that my high school physics teacher Mr. Bader had given me. One day he told me to stay after class. "Feynman," he said, "you talk too much and you make too much noise. I know why. You're bored. So I'm going to give you a book. You go up there in the back, in the corner, and study this book, and when you know everything that's in this book, you can talk again." So every physics class, I paid no attention to what was going on with Pascal's Law, or whatever they were doing. I was up in the back with this book: "Advanced Calculus", by Woods. Bader knew I had studied "Calculus for the Practical Man" a little bit, so he gave me the real works—it was for a junior or senior course in college. It had Fourier series, Bessel functions, determinants, elliptic functions—all kinds of wonderful stuff that I didn't know anything about. That book also showed how to differentiate parameters under the integral sign—it's a certain operation. It turns out that's not taught very much in the universities; they don't emphasize it. But I caught on how to use that method, and I used that one damn tool again and again. So because I was self-taught using that book, I had peculiar methods of doing integrals. The result was, when guys at MIT or Princeton had trouble doing a certain integral, it was because they couldn't do it with the standard methods they had learned in school. If it was contour integration, they would have found it; if it was a simple series expansion, they would have found it. Then I come along and try differentiating under the integral sign, and often it worked. So I got a great reputation for doing integrals, only because my box of tools was different from everybody else's, and they had tried all their tools on it before giving the problem to me.

See also[edit]

References[edit]

  1. ^ Protter, Murray H.; Morrey, Charles B., Jr. (1985). "Differentiation under the Integral Sign". Intermediate Calculus (Second ed.). New York: Springer. pp. 421–426. ISBN 0-387-96058-9. 
  2. ^ a b Flanders, Harly (June–July 1973). "Differentiation under the integral sign" (PDF). American Mathematical Monthly. 80 (6): 615–627. JSTOR 2319163. doi:10.2307/2319163. 

Further reading[edit]

  • Frederick S. Woods (1934). Advanced Calculus (New ed.). Ginn and Company. ASIN B0006AMNBI. 
  • Frederick S. Woods (1926). Advanced Calculus (1st ed.). Ginn and Company. ASIN B00085L67S. 
  • David V. Widder (Jul 1990). Advanced Calculus (New ed.). Dover Publications Inc. ISBN 978-0-486-66103-2. 

External links[edit]