- for .
This formula is the general form of the Leibniz integral rule and can be derived using the
fundamental theorem of calculus. The fundamental theorem of calculus is just a particular case of the above formula, for , a constant, and .
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)
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 connected region of and its boundary, respectively, is the Eulerian velocity at the boundary (see Lagrangian and Eulerian coordinates) and is unit outwards normal.
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 -dimensions is:
where is a time-varying domain of integration, is a -form, is the vector field of the velocity, , denotes the interior product, is the exterior derivative of with respect to the space variables only and is the time-derivative of .
Derivation of the principle of differentiation under the integral sign
A definite integral is a function of its upper limit and its lower limit .
If is a continuous function of or , then, from the definition of the definite integral, ,
Suppose and are constant, and that involves a parameter which is constant in the integration but may vary to form different integrals. Then, by the definition of a function,
In general, this may be differentiated by differentiating under the integral sign; i.e.,
To prove this and, at the same time, to determine conditions under which the formula is true, we proceed as follows:
From the fact that , we have
If is a continuous function of and when and lies between two values and , then may be taken to be so small that