Leibniz integral rule
Whether Leibniz's integral rule applies is essentially a question about the interchange of limits.
Stronger versions of the theorem only require that the partial derivative exist almost everywhere, and not that it be continuous.
[2] This formula is the general form of the Leibniz integral rule and can be derived using the fundamental theorem of calculus.
If both upper and lower limits are taken as constants, then the formula takes the shape of an operator equation:
In two and three dimensions, this rule is better known from the field of fluid dynamics as the Reynolds transport theorem:
is a scalar function, D(t) and ∂D(t) denote a time-varying connected region of R3 and its boundary, respectively,
The general statement of the Leibniz integral rule requires concepts from differential geometry, specifically differential forms, exterior derivatives, wedge products and interior products.
, dxω is the exterior derivative of ω with respect to the space variables only and
The above formula can be deduced directly from the fact that the Lie derivative interacts nicely with integration of differential forms
has only spatial components, the Lie derivative can be simplified using Cartan's magic formula, to
For t fixed, the mean value theorem implies there exists z in the interval [x, x + δ] such that
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.
and the dominated convergence theorem allows us to move the limit inside of the integral.
are all differentiable (see the remark at the end of the proof), by the multivariable chain rule, it follows that
There is a technical point in the proof above which is worth noting: applying the Chain Rule to
At time t the surface Σ in Figure 1 contains a set of points arranged about a centroid
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:
Having found the derivative, variables can be switched back to the original frame of reference.
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.
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.
An example of an application is the fact that power series are differentiable in their radius of convergence.
[citation needed] The Leibniz integral rule is used in the derivation of the Euler-Lagrange equation in variational calculus.
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 this technique, Feynman was able 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.
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.
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.
