An Introduction to Tensors and Group Theory for Physicists by Nadir Jeevanjee (2011) shows that this is a tensor that represents the volume of a parallelepiped.
It takes as input three vectors, and outputs one real number, the volume. And it is linear on each vector. This perfectly satisfied the definition of a tensor of order (3,0).
- math.stackexchange.com/questions/23312/what-is-the-importance-of-eigenvalues-eigenvectors/3503875#3503875
- math.stackexchange.com/questions/1520832/real-life-examples-for-eigenvalues-eigenvectors
- matheducators.stackexchange.com/questions/520/what-is-a-good-motivation-showcase-for-a-student-for-the-study-of-eigenvalues
Matrix congruence can be seen as the change of basis of a bilinear form by 
Ciro Santilli 40 Updated 2025-07-16
Ciro Santilli 40 Updated 2025-07-16From effect of a change of basis on the matrix of a bilinear form, remember that a change of basis modifies the matrix representation of a bilinear form as:
So, by taking , we understand that two matrices being congruent means that they can both correspond to the same bilinear form in different bases.
The different only shows up for field, not with particles. For fields, there are two types of changes that we can make that can keep the Lagrangian unchanged as mentioned at Physics from Symmetry by Jakob Schwichtenberg (2015) chapter "4.5.2 Noether's Theorem for Field Theories - Spacetime":
- spacetime symmetry: act with the Poincaré group on the Four-vector spacetime inputs of the field itself, i.e. transforming into
- internal symmetry: act on the output of the field, i.e.:
From the spacetime theory alone, we can derive the Lagrangian for the free theories for each spin:Then the internal symmetries are what add the interaction part of the Lagrangian, which then completes the Standard Model Lagrangian.
Node.js does have Node.js
worker_threads however. Quantum field theory lecture by Tobias Osborne (2017) Lecture 1 by Ciro Santilli 40 Updated 2025-07-16
Ciro Santilli 40 Updated 2025-07-16Bibliography review:
- Quantum Field Theory lecture notes by David Tong (2007) is the course basis
- quantum field theory in a nutshell by Anthony Zee (2010) is a good quick and dirty book to start
Course outline given:
- classical field theory
- quantum scalar field. Covers bosons, and is simpler to get intuition about.
- quantum Dirac field. Covers fermions
- interacting fields
- perturbation theory
- renormalization
Non-relativistic QFT is a limit of relativistic QFT, and can be used to describe for example condensed matter physics systems at very low temperature. But it is still very hard to make accurate measurements even in those experiments.
Mentions that "QFT is hard" because (a finite list follows???):But I guess that if you fully understand what that means precisely, QTF won't be too hard for you!
There are no nontrivial finite-dimensional unitary representations of the Poincaré group.
Notably, this is stark contrast with rotation symmetry groups (SO(3)) which appears in space rotations present in non-relativistic quantum mechanics.
www.youtube.com/watch?v=T58H6ofIOpE&t=5097 describes the relativistic particle in a box thought experiment with shrinking walls
Looks interesting.
It seems to abstract the part about the client messaging the backend, which focuses on being able to easily plug in a number of Front-end web framework to manage client state.
Uses Socket.IO, which allows the client Javascript to register callbacks when data is updated to achieve Socket.IO, e.g. their default chat app does:so that message appear immediately as they are sent.
client.service('messages').on('created', addMessage);Their standard template from But it is in itself a completely boring app with a single splash page, and no database interaction, so not a good showcase. The actual showcase app is feathersjs/feathers-chat.
feathers generate app on @feathersjs/cli@4.5.0 includes:which looks promising! They don't have a default template for a Front-end web framework however unfortunately: docs.feathersjs.com/guides/frameworks.html#the-feathers-chat lists a few chat app versions, which is their hello world:
- Front-end web framework: not built-in on generator, but there are some sample repos pointed from the documentation, and they did work out-of-box:
And there is no official example of the chat app that is immediately deployable to Heroku: FeathersJS Heroku deployment, all setups require thinking.
Tried a quick port to SQLite to get rid of annoying local databases for development, but failed, at c1c2cc4e448b279ff083272df1ac50d20c3304faandthen:fails with:Attempt to hack it:and after that it seems to run.
npm install sqlite3 --save-dev{
"type": "sqlite",
"database": "db.sqlite3",
"entities": ["src/**/**.entity{.ts,.js}"],
"synchronize": true
}npm startDataTypeNotSupportedError: Data type "timestamp" in "ArticleEntity.created" is not supported by "sqlite" database.--- a/src/article/article.entity.ts
+++ b/src/article/article.entity.ts
@@ -20,10 +20,10 @@ export class ArticleEntity {
@Column({default: ''})
body: string;
- @Column({ type: 'timestamp', default: () => "CURRENT_TIMESTAMP"})
+ @Column({ default: () => "CURRENT_TIMESTAMP"})
created: Date;
- @Column({ type: 'timestamp', default: () => "CURRENT_TIMESTAMP"})
+ @Column({ default: () => "CURRENT_TIMESTAMP"})
updated: Date;I can signup and login, terrible error reporting as usual, make sure to use long enough usernames/passwords.
However, article creation fails with:
Unhandled Rejection (TypeError): Cannot read property 'slug' of undefined 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!
Intro to OurBigBook
. Source. We have two killer features:
- 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-calculusArticles 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/derivativeVideo 2. OurBigBook Web topics demo. Source. - 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.
- to OurBigBook.com to get awesome multi-user features like topics and likes
- as HTML files to a static website, which you can host yourself for free on many external providers like GitHub Pages, and remain in full control
Figure 3. Visual Studio Code extension installation.
Figure 4. Visual Studio Code extension tree navigation.
Figure 5. Web editor. 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.Video 4. OurBigBook Visual Studio Code extension editing and navigation demo. Source. 
- Infinitely deep tables of contents:
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