This has been something that always seems a bit magical to me, when someone comes up with an obscure change of variables that completely simplifies the problem. I'd love to learn this in a more structured/rigorous way. Does anyone have any recommendations for picking this up?
I think there is less of unifying theory than an assortment of artful tricks - and it's a pretty small assortment, too.
If you want to be able to reproduce every clever mathematical trick of the last 200 years you probably can (I believe in you!). But everyone short of Ramanujan learns them by studying many problems, repeated exposure, and either systematically or organically memorizing them. Afterwards, using them in a problem is more about pattern recognition. This is my experience as a physics phd-master-out ymmv
People might disagree strongly, but my recommendation for you is to look into lambda calculus for this.
There are two important concepts that can be generalized:
- Substitution, the method of renaming and replacing variables
- Abstraction, replacing a variable with another term (function) that can also serve as output
This should give you a more natural feel to concepts like parametrization or what it means that the solution to one function may be another function. Perhaps you might even look a bit into the simply typed lambda calculus which has given me yet more perspectives on the fundamentals of mathematics.
This has been something that always seems a bit magical to me, when someone comes up with an obscure change of variables that completely simplifies the problem. I'd love to learn this in a more structured/rigorous way. Does anyone have any recommendations for picking this up?