Ciro Santilli @cirosantilli  40

Whenever you make a change to your material, people should still be able to access the previous version.

Maybe there was something in the previous version that they needed, and you just removed.

Git + GitHub is the perfect way to do versioning.

Keep the example/theory ratio high, very, very high.

For natural sciences, add as many reproducible experiment images/videos/descriptions as you can.

And if you really can't make money from a subject, there is only one other thing people crave: beauty.

You have to give the beauty motivations upfront, before boring people to death with endless prerequisites, otherwise no one will ever want to learn it.

The thing about projects is that they are illiquid: it is not easy to immediately compare them.

And that is the whole point.

The outcome of that however is that you have to learn how to explain what you've achieved to others and why it is awesome.

Just like in the real world.

You have to create portfolio, and do some public relations.

This is the only way to truly understand and appreciate the subject.

Understanding the experiments gets intimately entangled with basically learning the history of physics, which is extremely beneficial as also highlighted by Ron Maimon, related: there is value in tutorials written by early pioneers of the field.

"How we know" is a basically more fundamental point than "what we know" in the natural sciences.

In the Surely You're Joking, Mr. Feynman chapter O Americano, Outra Vez! Richard Feynman describes his experience teaching in Brazil in the early 1950s, and how everything was memorized, without any explanation of the experiments or that the theory has some relationship to the real world!

Although things have improved considerably since in Brazil, Ciro still feels that some areas of physics are still taught without enough experiments described upfront. Notably, ironically, quantum field theory, which is where Feynman himself worked.

Feynman gave huge importance to understanding and explaining experiments, as can also be seen on Richard Feynman Quantum Electrodynamics Lecture at University of Auckland (1979).

In Physics, in order to test a theory, you must be able to extract a number from it.

It does not matter how, if it is exact, or numerical, or a message from God: a number has to come out of the formulas in the end, and you have to compare it with the experimental data.

Many theoretical physicists seem to forget this in their lectures, see also: Section "How to teach and learn physics".

Nature is a black box, right?

You don't need to understand the from first principles derivation of every single phenomena.

And most important of all: you should not start learning phenomena by reading the from first principles derivation.

Instead, you should see what happens in experiments, and how matches some known formula (which hopefully has been derived from first principles).

Only open the boxes (understand from first principles derivation) if the need is felt!

E.g.:

Physics is all about predicting the future. If you can predict the future with an end result, that's already predicting the future, and valid.

There are two cases:

Questions: are all compact manifolds / differential manifolds homotopic / diffeomorphic to the sphere in that dimension?

www.youtube.com/watch?v=0hfdSrK46EE How To Use A Minimoog Synthesizer by Doctor Mix (2020)

Software emulators:

See also some remarks of Ciro Santilli's thoughts on the instrument Ciro Santilli's musical education.

Something analogous to a group isomorphism, but that preserves whatever properties the given algebraic object has. E.g. for a field, we also have to preserve multiplication in addition to addition.

Other common examples include isomorphisms of vector spaces and field. But since both of those two are much simpler than groups in classification, as they are both determined by number of elements/dimension alone, see:we tend to not talk about isomorphisms so much in those contexts.

Like isomorphism, but does not have to be one-to-one: multiple different inputs can have the same output.

The image is as for any function smaller or equal in size as the domain of course.

This brings us to the key intuition about group homomorphisms: they are a way to split out a larger group into smaller groups that retains a subset of the original structure.

As shown by the fundamental theorem on homomorphisms, each group homomorphism is fully characterized by a normal subgroup of the domain.