A gauge transformation is
Because partial derivatives commute, the added contribution to is
so the electromagnetic field tensor is gauge invariant.
Define
Since and ,
The identity
follows directly from . Its spatial and mixed components are
For the other two equations define the four-current
Then gives
and gives
This is the Covariant Maxwell equation with the minus-plus-plus-plus metric.
Using
one finds
This is the electromagnetic energy density.
For a null vector, the trace term in vanishes. Put
Antisymmetry gives . A vector orthogonal to a null vector has nonnegative Minkowski norm, so
Explicitly, in a frame with ,
Hence the null energy condition for the electromagnetic field is strict whenever the contraction is nonzero:
The convolution is
We claim that the -fold convolution is
This is true for . If it holds for and , then
and the convolution vanishes for . This is the gamma density from repeated exponential convolution.
The Fourier transform is
The convolution theorem states
Indeed, Fubini and give
Since , induction immediately verifies
For Parseval identity, take . Then
Using Fourier inversion at zero, which follows from the supplied delta identity,
and the convolution theorem gives
Apply this to . Since
one obtains the Even rational Parseval integral
For a variation ,
Integration by parts gives
For fixed endpoints, . The fundamental lemma of the calculus of variations therefore gives the Euler-Lagrange equation
A solution makes the first variation vanish for every admissible variation, so it is a stationary candidate; whether it is a minimum or maximum is decided by higher variations. If endpoint values are free, is arbitrary there, and the boundary term instead vanishes under the natural conditions
Thus the same Euler-Lagrange solution is stationary for all free-endpoint variations. These are the natural boundary conditions for a free endpoint.
For
the two equations are
Set and . Then
The conditions give
and hence the most general solution is
Free conditions at are . Adding and subtracting them gives
Thus . The zero solution exists for every , while nonzero solutions exist precisely when
For those values they form the one-parameter family
where is arbitrary. This is the free-endpoint normal mode of a coupled variational functional.
For , the beta-integral evaluation gives
Differentiation under the integral sign near is justified by domination at zero and infinity. Therefore
Writing gives
Since
the logarithmic moments of the Cauchy kernel yield
The same expansion gives , agreeing with the supplied identity.

Pinned article: Introduction to the OurBigBook Project

Welcome to the OurBigBook Project! Our goal is to create the perfect publishing platform for STEM subjects, and get university-level students to write the best free STEM tutorials ever.
Everyone is welcome to create an account and play with the site: ourbigbook.com/go/register. We belive that students themselves can write amazing tutorials, but teachers are welcome too. You can write about anything you want, it doesn't have to be STEM or even educational. Silly test content is very welcome and you won't be penalized in any way. Just keep it legal!
We have two killer features:
  1. topics: topics group articles by different users with the same title, e.g. here is the topic for the "Fundamental Theorem of Calculus" ourbigbook.com/go/topic/fundamental-theorem-of-calculus
    Articles of different users are sorted by upvote within each article page. This feature is a bit like:
    • a Wikipedia where each user can have their own version of each article
    • a Q&A website like Stack Overflow, where multiple people can give their views on a given topic, and the best ones are sorted by upvote. Except you don't need to wait for someone to ask first, and any topic goes, no matter how narrow or broad
    This feature makes it possible for readers to find better explanations of any topic created by other writers. And it allows writers to create an explanation in a place that readers might actually find it.
    Figure 1.
    Screenshot of the "Derivative" topic page
    . View it live at: ourbigbook.com/go/topic/derivative
  2. local editing: you can store all your personal knowledge base content locally in a plaintext markup format that can be edited locally and published either:
    This way you can be sure that even if OurBigBook.com were to go down one day (which we have no plans to do as it is quite cheap to host!), your content will still be perfectly readable as a static site.
    Figure 5. . You can also edit articles on the Web editor without installing anything locally.
    Video 3.
    Edit locally and publish demo
    . Source. This shows editing OurBigBook Markup and publishing it using the Visual Studio Code extension.
  3. https://raw.githubusercontent.com/ourbigbook/ourbigbook-media/master/feature/x/hilbert-space-arrow.png
  4. Infinitely deep tables of contents:
    Figure 6.
    Dynamic article tree with infinitely deep table of contents
    .
    Descendant pages can also show up as toplevel e.g.: ourbigbook.com/cirosantilli/chordate-subclade
All our software is open source and hosted at: github.com/ourbigbook/ourbigbook
Further documentation can be found at: docs.ourbigbook.com
Feel free to reach our to us for any help or suggestions: docs.ourbigbook.com/#contact