Writing the velocity as gives
so the flow is incompressible. With the convention
a stream function is
The streamlines are its level sets. For , they are the hyperbolas
with separatrices and a saddle at the origin. For , they are concentric ellipses
traversed clockwise. This is the streamline classification of a planar linear saddle or centre.
The scalar vorticity is
Hence the flow is irrotational exactly when
Then , and a velocity potential is
Vanishing net charge density does not require vanishing current. Positive and negative charge carriers can cancel in charge density while their oppositely directed motions add to a nonzero current. Charge conservation only requires
for magnetostatics this becomes .
Because , one may introduce a magnetic vector potential with
It is not unique: gives the same field for any scalar .
For the stated current,
so it is consistent with stationary charge conservation. Direct calculation gives
Thus is a Beltrami field. For , choose
Then and . A convenient Coulomb-gauge potential is
since and . Gradient gauge terms may of course be added.
If , the current is the constant field . The formulas involving do not apply; one valid choice is
Multiplication by puts the equation in Sturm-Liouville theory form:
Multiply the equations for by , subtract and integrate. The boundary term vanishes because at both endpoints and the polynomial derivatives are bounded. Therefore, for ,
This is the usual Chebyshev polynomial orthogonality.
Differentiating the original equation and writing gives
Its self-adjoint form is
The same subtraction argument now yields, for ,
These two relations form the Chebyshev derivative Sturm-Liouville pair.
A map is a contraction if there is a constant such that
for every .
The contraction mapping theorem states that a contraction of a nonempty complete metric space has a unique fixed point, and that the iterates from every starting point converge to it. To prove this, choose and put . Then
so, for ,
Thus is Cauchy and converges, by completeness, to some . A contraction is continuous, so
If is another fixed point, then
forcing .
For the Newton map
one has and
On the given neighbourhood,
Since , choose a closed interval centred at and contained in so small that there. Then
so and is a contraction on the complete interval . The local contraction proof for Newton iteration therefore shows that is the unique fixed point of on .
The Eisenstein criterion says that a primitive polynomial
is irreducible if there is a prime such that
By the Gauss lemma for polynomials, irreducibility over and over agree for primitive polynomials.
If the urn contains green balls, it contains red balls: both update rules preserve the difference . Until absorption at , the green count is therefore a birth-death chain with
The function
is harmonic, since
Let and be the hitting times of and . The optional sampling theorem for a supermartingale, applied to the bounded stopped martingale , gives
Solving,
Letting , the events on the left increase to eventual termination. The harmonic hitting probability for the balanced-difference urn is therefore
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:

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