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Stem Formulas

stemformulas.com

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Hi everyone, I'm the creator of this! Thanks for posting this, I tried to post it once and it just didn't take off so I wasn't sure if I should post again. I'll read all the feedback now.

Cool! I've just contributed several examples. If anyone is interested in the sheer amount of identities that have been discovered, good books are (many of them gigantic references spanning thousands of pages). When bored, try proving some of those facts, examples build on top of each other. These are not the only examples, as there are many texts like these in other areas of mathematics and engineering, be it numerical analysis, optimization and variational analysis, statistics, abstract algebra, control theory, geometry and so on.

Table of Integrals, Series, and Products, Gradshteyn & Ryzhik.

Special Integrals of Gradshteyn and Ryzhik, Vols. I and II, Moll for some proofs of the above.

Handbook of Integral Equations, Polyanin & Manzhirov.

Scalar, Vector, and Matrix Mathematics, Bernstein.

Handbook of Number Theory I and II, Sandor, Crstici & Mitrinovic.

Wikipedia also has a plethora of pages with mathematical identities. Some of them:

https://en.wikipedia.org/wiki/Vector_calculus_identities

https://en.wikipedia.org/wiki/Vector_algebra_relations

https://en.wikipedia.org/wiki/Exterior_calculus_identities

https://en.wikipedia.org/wiki/Del_in_cylindrical_and_spheric...

https://en.wikipedia.org/wiki/List_of_formulas_in_Riemannian...

https://en.wikipedia.org/wiki/List_of_set_identities_and_rel...

https://en.wikipedia.org/wiki/List_of_triangle_inequalities

https://en.wikipedia.org/wiki/List_of_trigonometric_identiti...

... and its several lists of integrals (including trigonometric, exponential, rational).

https://en.wikipedia.org/wiki/Lists_of_integrals#Lists_of_in...

More advanced topics:

http://proximity-operator.net/

Can't have a list like that without a mention of Abramowitz & Stegun [0], or its successor, the NIST Digital Library of Mathematical Functions [1]. It's about as comprehensive as it gets.

[0] https://en.wikipedia.org/wiki/Abramowitz_and_Stegun

[1] https://dlmf.nist.gov/

I'm still holding on to my CRC Standard Mathematical Tables and Abramowitz and Stegun for when they turn the Internet off.

Clean!

I submitted the first formula that popped into my head from uni years ago: the Generalised Linear Model definition from one of the later stats courses

This project vaguely reminds me of Gradshteyn and Ryzhik (https://en.wikipedia.org/wiki/Gradshteyn_and_Ryzhik)

There's an effort to provide proofs for every single integral in Gradshteyn & Ryzhik.

http://www.math.tulane.edu/~vhm/Table.html

Rosetta Code: https://rosettacode.org/

> Complexity Zoo > Petting Zoo > {P, NP, PP,}, Modeling Computation > Deterministic Turing Machine https://complexityzoo.net/Petting_Zoo#Deterministic_Turing_M...

From https://news.ycombinator.com/item?id=31719696 :

> Fixed-point combinator > Y Combinator, Implementations in other languages: https://en.wikipedia.org/wiki/Fixed-point_combinator

> Y-combinator in like 100 languages: https://rosettacode.org/wiki/Y_combinator #Python

Quantum Algorithm Zoo by Microsoft Quantum: https://quantumalgorithmzoo.org/

From https://westurner.github.io/hnlog/#comment-30784573 :

> Quantum Monte Carlo,

QFT and iQFT; Inverted Quantum Fourier Transform: https://en.wikipedia.org/wiki/Quantum_Fourier_transform

From "Common Lisp names all sixteen binary logic gates" https://news.ycombinator.com/item?id=32804463 :

- Quantum_logic

The matrix representations of quantum gates and operators are neat too; https://www.google.com/search?q=matrix+representations+of+qu...

Those formulas with i as interest rates are really tripping me up. I mean (1+i)^n just looks like it aught to be complex.

Edit: I'm also not too happy with the explanation of Feymann's trick, it fails to properly explain Feynman's trick, says n! is the gamma function, and applies the trick to an integral which follows directly from the definition of the Gamma function and cannot be simplified without that definition.

Good news! Real numbers are complex numbers!

That’s accidental complexity though, not essential complexity. ;)

The "i" in (1+i)^n is the interest rate.

Hi, creator here. Thanks for the feedback, I've been kind of using this site just to store things that I find personally useful and in ways that I, an undergraduate student, understand so I'm sure there are some corners being cut like in the case of Feynman's trick. Do you think you could help me improve that page? The source code is here: https://github.com/stemformulas/stemformulas.github.io/blob/...

I'd remove that page entirely, perhaps in favour of one that simply lists the definition of the Gamma function (which is basically the formula there right now).

You can't summarize Feynman's trick in one formula. At best you can list an example, and I'd argue this isn't the best one.

Great site! Definitely should not be paginated though. Just put them all on one page.

Is this HN's auto-capitalizer? The title should be “STEM Formulas”.

It's a nice collection, it could keep me glued for hours thinking about all this stuff.

Yes (probably). If you notice this kind of errors, you can send an email to the mods hn@ycombinator.com so they fix it manualy.

I think this is great, though still only about 80% there.

Some helpful additions would be labeling which values are constants, examples of both how and where the equation is used, and numerical representations of the formulae.

When I click on Schrodinger's eq I want to be able to click on the Wave function and see an example of the numerical form, ie a matrix of vectors with toy values.

When I started out, I added examples to formulas, but it did require a lot more effort per formula and as I am adding these formulas on my own (contributors please!) I chose to prioritize quantity first. The rest of your suggestions are good, I'll add them to my list of ideas, thank you.

Recent and related:

Show HN: Stemformulas.com - https://news.ycombinator.com/item?id=36737925 - July 2023 (3 comments)

(reposts are fine if an article didn't get significant attention - this one is just interesting because it's a Show HN)

Some of those are just definitions. Not sure what's the usefulness of det(A) = |A| or c=λf, it's not like one would need to look that up.

The first one just looks like a tautology but the latter might be helpful, even if it is a bit trivial if you think about it.

Clicking on the det card shows definitions for 2x2 and 3x3 matrices, probably wouldnt fit on the card

Formulas are like magic spells. You don’t get to use them without understanding what they mean, and more importantly what they assume.

The matrix formula is one of my favourites because it works for any ring, including the case where abcd elements are matrices themselves.

Looks like a worse version of ProofWiki but with a more modern design.

Yeah, it's gonna be a while until I can have a lot of formulas, but I do think I'm adding value in a few ways:

- Copying LaTeX on formula pages - Formulas have an open-graph preview so if they are linked on social media you can see the formulas in the preview - The search is the main page, in focus, so you can realistically open a new browser tab and find a formula that you know exists in under a second. - Hidden feature: the forward-slash (/) key also triggers the search dialogue, so minimal mouse is needed ever while navigating the site.

Is shocking for me how after studying Telco Eng. I've used or learnt about 90% of them, and it's sad how most of us would never use them again.

Nice layout, but I still prefer Wikipedia.

This could be seen as an index of wikipedia formulas (each formula links to wikipedia, anyway).

I built this because of my frustration with Wikipedia actually. A lot of Wikipedia pages are really rigorous, but when I visit I just want to know the common form of the main equation, which is sometimes placed a few headings down.

Could anyone share an advice or resources where to start with learning to read mathematical notations?

Lean mathlib may already have the proofs for many of these Stem Formulas (in LaTeX)? These formulas as SymPy would also be useful.

latex2sympy parses LaTeX and generates SymPy symbolic CAS Python code (w/ ANTLR) and is now merged in SymPy core but you must install ANTLR before because it's an optional dependency. Then, sympy.lambdify will compile a symbolic expression for use with TODO JAX, TensorFlow, PyTorch,.

"Ask HN: Did studying proof based math topics make you a better programmer?" Re: lean mathlib https://news.ycombinator.com/item?id=36463580

[Mathematics in mathlib > A mathlib overview]( https://leanprover-community.github.io/mathlib-overview.html )

From https://news.ycombinator.com/item?id=36159017 :

> sympy.utilities.lambdify.lambdify() https://github.com/sympy/sympy/blob/a76b02fcd3a8b7f79b3a88df... :

>> """Convert a SymPy expression into a function that allows for fast numeric evaluation [with the CPython math module, mpmath, NumPy, SciPy, CuPy, JAX, TensorFlow, SymPy, numexpr,]*

From https://westurner.github.io/hnlog/#comment-19084622 :

> "latex2sympy parses LaTeX math expressions and converts it into the equivalent SymPy form" and is now merged into SymPy master and callable with sympy.parsing.latex.parse_latex(). It requires antlr-python-runtime to be installed. https://github.com/augustt198/latex2sympy https://github.com/sympy/sympy/pull/13706

ENH: 'generate a Jupyter notebook' (nbformat .ipynb JSON) function from this stem formula

ENH: Store/export Stem formula attributes as JSON-LD Linked Data and/or RDFa (RDF in HTML Attributes) .

JSON-LD Playground has examples of JSON-LD, as does https://schema.org/CreativeWork : https://json-ld.org/playground/

Pending are the schema.org RDFS vocabulary MathSolver class and mathExpression property: https://schema.org/MathSolver and https://schema.org/mathExpression

Dbpedia has wikipedia infobox attributes as RDF.

IDK if there's an RDF interface to Wolfram?

ENH: Generate search urls from formula names

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