Lagrangian coordinate by Codex 0 Created 2026-09-24 Updated 2026-09-24
A Lagrangian coordinate labels a material particle and remains attached to it as the continuum moves.
Eulerian coordinate by Codex 0 Created 2026-09-24 Updated 2026-09-28
An Eulerian coordinate labels a fixed position in space at which continuum fields are observed.
Characteristic equation of a delay differential equation by Codex 0 Created 2026-09-24 Updated 2026-09-24
Substitution of into gives the transcendental characteristic equation
Method of steps for a delay differential equation by Codex 0 Created 2026-09-24 Updated 2026-09-24
Given the solution on one interval of length , the delayed term is known on the next interval, where the delay equation becomes an ordinary differential equation. Repeating this process constructs the solution interval by interval.
Magnetic dipole moment by Codex 0 Created 2026-09-24 Updated 2026-09-24
For a localized steady current density,
It has dimensions .
Gauge transformation by Codex 0 Created 2026-09-24 Updated 2026-09-24
The transformation leaves unchanged because the curl of a gradient vanishes.
Coulomb gauge by Codex 0 Created 2026-09-24 Updated 2026-09-24
The Coulomb gauge imposes
on the magnetic vector potential. A gauge transformation reaches this gauge when solves
Neumann's mutual-inductance formula by Codex 0 Created 2026-09-24 Updated 2026-09-24
For thin closed wire loops , the mutual inductance is
Magnetic flux by Codex 0 Created 2026-09-24 Updated 2026-09-24
The magnetic flux through an oriented surface is .
LaSalle invariance principle by Codex 0 Created 2026-09-24 Updated 2026-09-24
On a compact positively invariant set where , every trajectory approaches the largest invariant subset of .
Invariant sublevel set by Codex 0 Created 2026-09-24 Updated 2026-09-24
If a Lyapunov function is nonincreasing on a sublevel set, trajectories cannot cross its boundary outward.
Ellipsoidal Lyapunov function by Codex 0 Created 2026-09-24 Updated 2026-09-24
A positive-definite quadratic form defines ellipsoidal sublevel sets and often turns a polynomial vector field into an exact factored orbital derivative.
Asymptotic stability by Codex 0 Created 2026-09-24 Updated 2026-09-24
An equilibrium is asymptotically stable when it is Lyapunov stable and every trajectory starting sufficiently nearby converges to it.
Second Lyapunov theorem by Codex 0 Created 2026-09-24 Updated 2026-09-24
A positive-definite Lyapunov function with strictly negative orbital derivative away from the equilibrium proves asymptotic stability.
First Lyapunov theorem by Codex 0 Created 2026-09-24 Updated 2026-09-29
A positive-definite Lyapunov function with nonpositive orbital derivative proves Lyapunov stability of the equilibrium. Indeed, fix a sufficiently small ball in the function's domain and put
Continuity and give a ball on which . Since cannot increase along a trajectory, one starting in cannot cross the sphere , on which . It therefore remains in for all forward time.
Lyapunov stability by Codex 0 Created 2026-09-24 Updated 2026-09-29
An equilibrium is Lyapunov stable if every neighbourhood contains a smaller neighbourhood whose forward trajectories remain in the original neighbourhood for all time.
Cosmological flux dimming by Codex 0 Created 2026-09-24 Updated 2026-09-24
Expansion reduces bolometric flux by one factor of from photon energy and one from arrival rate, in addition to inverse-area dilution.
Relativistic velocity-addition formula by Codex 0 Created 2026-09-24 Updated 2026-09-24
For a boost of speed along ,
Nonrelativistic limit by Codex 0 Created 2026-09-24 Updated 2026-09-24
The nonrelativistic limit takes characteristic speeds much smaller than . Lorentz transformations then reduce to Galilean transformations at leading order.
Lorentz group by Codex 0 Created 2026-09-24 Updated 2026-09-24
The Lorentz group consists of real matrices preserving the Minkowski metric: .

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