In calculus, the Leibniz integral rule or the Leibniz rule for differentiation under the integral sign, named after Gottfried Wilhelm Leibniz, states that for an integral of the form where and the integrands are functions dependent on the derivative of this integral is expressible as
where the partial derivative indicates that inside the integral, only the variation of with is considered in taking the derivative.
In the special case where the functions and are constants and with values that do not depend on this simplifies to:
If is constant and , which is another common situation (for example, in the proof of Cauchy's repeated integration formula), the Leibniz integral rule becomes:
This important result may, under certain conditions, be used to interchange the integral and partial differential operators, and 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
Intuitively, Leibniz's Integral rule says that if you have a definite integral moving in time, the rate of change of the integral is the sum of the rate of change of the integrand and the difference in the fluxes at the boundaries of the integral. Formally,
The right hand side may also be written using Lagrange's notation as:
Stronger versions of the theorem only require that the partial derivative exist almost everywhere, and not that it be continuous. 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 is constant, and does not depend on
If both upper and lower limits are taken as constants, then the formula takes the shape of an operator equation:
where is the partial derivative with respect to and is the integral operator with respect to 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
A Leibniz integral rule for a two dimensional surface moving in three dimensional space is
where: is a vector field at the spatial position at time , is the partial time derivative of the vector field, is a surface bounded by the closed curve , is a vector element of the surface , is a vector element of the curve , is the velocity of movement of the region , is the vector divergence, is the vector cross product, The double integrals are surface integrals over the surface , and the line integral is over the bounding curve .
Higher dimensions
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, and 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 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 n dimensions is
where is a time-varying domain of integration, ω is a p-form, is the vector field of the velocity, denotes the interior product with , dxω is the exterior derivative of ω with respect to the space variables only and is the time derivative of ω.
The above formula can be deduced directly from the fact that the Lie derivative interacts nicely with integration of differential forms
for the spacetime manifold , where the spacetime exterior derivative of is and the surface has spacetime velocity field . Since has only spatial components, the Lie derivative can be simplified using Cartan's magic formula, to
which, after integrating over and using generalized Stokes' theorem on the second term, reduces to the three desired terms.
Measure theory statement
Let be an open subset of , and be a measure space. Suppose satisfies the following conditions:
is a Lebesgue-integrable function of for each . For almost all , the partial derivative exists for all . There is an integrable function such that for all and almost every .
Then, for all ,
The proof relies on the dominated convergence theorem and the mean value theorem (details below).
Proofs
Proof of basic form We first prove the case of constant limits of integration a and b.
We use Fubini's theorem to change the order of integration. For every and , such that and both and are within , we have:
Note that the integrals at hand are well defined since is continuous at the closed rectangle and thus also uniformly continuous there; thus its integrals by either dt or dx are continuous in the other variable and also integrable by it (essentially this is because for uniformly continuous functions, one may pass the limit through the integration sign, as elaborated below).
Therefore:
Where we have defined:
(we may replace x0 here by any other point between x0 and x)
F is differentiable with derivative , so we can take the limit where approaches zero. For the left hand side this limit is:
For the right hand side, we get:
And we thus prove the desired result:
Another proof using the bounded convergence theorem If the integrals at hand are Lebesgue integrals, we may use the bounded convergence theorem (valid for these integrals, but not for Riemann integrals) in order to show that the limit can be passed through the integral sign.
Note that this proof is weaker in the sense that it only shows that fx(x,t) is Lebesgue integrable, but not that it is Riemann integrable. In the former (stronger) proof, if f(x,t) is Riemann integrable, then so is fx(x,t) (and thus is obviously also Lebesgue integrable).
Let
By the definition of the derivative,
Substitute equation () into equation (). The difference of two integrals equals the integral of the difference, and 1/h is a constant, so
We now show that the limit can be passed through the integral sign.
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 ) 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.
If instead we only know that there is an integrable function such that , then and the dominated convergence theorem allows us to move the limit inside of the integral.
Variable limits form For a continuous real valued function g of one real variable, and real valued differentiable functions and of one real variable,
This follows from the chain rule and the First Fundamental Theorem of Calculus. Define
and
(The lower limit just has to be some number in the domain of )
Then, can be written as a composition: . The Chain Rule then implies that
By the First Fundamental Theorem of Calculus, . Therefore, substituting this result above, we get the desired equation:
Note: This form can be particularly useful if the expression to be differentiated is of the form:
Because does not depend on the limits of integration, it may be moved out from under the integral sign, and the above form may be used with the Product rule, i.e.,
General form with variable limits 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 ξ1 → a and ξ2 → b. 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,
Alternative proof of the general form with variable limits, using the chain rule
The general form of Leibniz's Integral Rule with variable limits can be derived as a consequence of the basic form of Leibniz's Integral Rule, the multivariable chain rule, and the first fundamental theorem of calculus. Suppose is defined in a rectangle in the - plane, for and . Also, assume and the partial derivative are both continuous functions on this rectangle. Suppose are differentiable real valued functions defined on , with values in (i.e. for every ). Now, set
and
Then, by properties of definite Integrals, we can write
Since the functions are all differentiable (see the remark at the end of the proof), by the multivariable chain rule, it follows that is differentiable, and its derivative is given by the formula: Now, note that for every , and for every , we have that , because when taking the partial derivative with respect to of , we are keeping fixed in the expression ; thus the basic form of Leibniz's Integral Rule with constant limits of integration applies. Next, by the first fundamental theorem of calculus, we have that ; because when taking the partial derivative with respect to of , the first variable is fixed, so the fundamental theorem can indeed be applied.
Substituting these results into the equation for above gives:
as desired.
There is a technical point in the proof above which is worth noting: applying the Chain Rule to requires that already be differentiable. This is where we use our assumptions about . As mentioned above, the partial derivatives of are given by the formulas and . Since is continuous, its integral is also a continuous function, and since is also continuous, these two results show that both the partial derivatives of are continuous. Since continuity of partial derivatives implies differentiability of the function, is indeed differentiable.
Three-dimensional, time-dependent form
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