Once you have crated something awesome, you have to advertise it, otherwise no one will ever find it.
This means:
- whenever you walk into a classroom, give students a link to the materialThen ask them if they want to talk about anything.Then leave the classroom and go produce more good material instead of wasting your time there :-)
- whenever someone asks as question on an online forum, answer it, and link to the section of your material that also answers that question.The material will answer many of their future questions.
- after you've done something awesome, Google possible relevant keywords that should hit it.This will lead you to other websites that talk about the same content.Then, leave comments on those pages linking to your stuff, or email the authors of those pages.It is borderline spam, but if the subject is closely related, it is a win for everyone.
Eventually, people will find you on the front page of Google searches, and then you will know that you've truly made something useful.
Talk with individuals, not to groups by 
Ciro Santilli 35 Updated 2025-03-28 +Created 1970-01-01
Ciro Santilli 35 Updated 2025-03-28 +Created 1970-01-01When you do get face to face time with students, don't teach a large group.
Everything you want to teach is already online.
And if it is not, then you are wasting your time saying it face-to-face instead of creating such online resource.
The only goal of meeting students is talking to them individually or in small groups to:
- understand what they feel
- transmit your passion for the subject
and letting them do the same amongst themselves.
If you talk to a large group, you will only reach / understand a very small percentage of the group, so your time is wasted.
It is better to deeply understand what 25% of the students feel and adapt the course material, than to talk to everyone at once, and have only 5% understand anything.
Finding a complete basis such that each vector solves a given differential equation is the basic method of solving partial differential equation through separation of variables.
The first example of this you must see is solving partial differential equations with the Fourier series.
Notable examples:
- Fourier series for the heat equation as shown at Fourier basis is complete for and solving partial differential equations with the Fourier series
- Hermite functions for the quantum harmonic oscillator
- Legendre polynomials for Laplace's equation in spherical coordinates
- Bessel function for the 2D wave equation on a circular domain in polar coordinates
Directly modelled by group.
For continuous symmetries, see: Lie group.
Takes a vector field as input and produces a scalar field.
Mnemonic: it gives out the amount of fluid that is going in or out of a given volume per unit of time.
Therefore, if you take a cubic volume:
- the input has to be the 6 flows across each face, therefore 3 derivatives
- the output is the variation of the quantity of fluid, and therefore a scalar
Ciro Santilli thinks that maybe the government does not need to provide those, but it needs to regulate the fuck out of them, notably control over censorship in those platforms: the deplatforming of Donald Trump.
Related:
This section is about functions that operates on arbitrary sets.
MOOCs are a bad idea. We don't want to simply map the pre-computer classroom to the Internet. The Internet allows, and requires, fundamentally new ways to do things. More like Stack Overflow/Wikipedia. More like OurBigBook.com.
Being naughty and creative are correlated by Ciro Santilli 35 Updated 2025-03-28 +Created 1970-01-01
Ciro Santilli 35 Updated 2025-03-28 +Created 1970-01-01Order of the highest derivative that appears.
This basically adds one more ingredient to partial differential equations: a function that we can select.
And then the question becomes: if this function has such and such limitation, can we make the solution of the differential equation have such and such property?
It's quite fun from a mathematics point of view!
Control theory also takes into consideration possible discretization of the domain, which allows using numerical methods to solve partial differential equations, as well as digital, rather than analogue control methods.
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