The easy and less generic integral. The harder one is the Lebesgue integral.
This is the most important of all points.
Don't set goals for your students.
Ask students what they want to do, and help them achieve that goal.
If they don't know what to do, give suggestions of interesting things they could do.
Once they have a goal, just help them learn everything that is needed to achieve that goal
If they don't have a goal, any attempt to learn is a total and complete waste of time.
This is because the universe of potentially useful things that can be learnt is infinite, and no human can ever learn everything.
The only solution, is to try and learn only what seems necessary to reach your goal, and just try to reach your goal instead.
This approach is called backward design.
Also, setting overly ambitious goals, is a good idea: the side effects of ambitious goals are often the most valuable thing achieved.
"Graduating" and "getting a diploma" are not valid goals, because they are useless. A goal has to be either an amazing specific technological or artistic development.
If you give a course in a classroom, you reach 10 people (the others were sleeping).
If you make a perfect course online, and answer questions online, you reach 10 thousand.
Not doing things online is a waste of time.
You are a highly trained professional, and your time is extremely valuable.
Even if it takes twice as long to create the material than giving course, you are still more efficient by a factor of 500.
It is as if there were 500 little copies of you working full time. It is a superpower.
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 2025-01-10 +Created 1970-01-01
Ciro Santilli 35 Updated 2025-01-10 +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 2025-01-10 +Created 1970-01-01
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