A fancy name for calculus, with the "more advanced" connotation.
A ring where multiplication is commutative and there is always an inverse.
A field can be seen as an Abelian group that has two group operations defined on it: addition and multiplication.
And then, besides each of the two operations obeying the group axioms individually, and they are compatible between themselves according to the distributive property.
Basically the nicest, least restrictive, 2-operation type of algebra.
Examples:
Main motivation: Lebesgue integral.
The Bright Side Of Mathematics 2019 playlist: www.youtube.com/watch?v=xZ69KEg7ccU&list=PLBh2i93oe2qvMVqAzsX1Kuv6-4fjazZ8j
The key idea, is that we can't define a measure for the power set of R. Rather, we must select a large measurable subset, and the Borel sigma algebra is a good choice that matches intuitions.
Where derivation == "intuitive routes", since a "law of physics" cannot be derived, only observed right or wrong.
TODO also comment on why are complex numbers used in the Schrodinger equation?.
Some approaches:
- en.wikipedia.org/w/index.php?title=Schr%C3%B6dinger_equation&oldid=964460597#Derivation: holy crap, this just goes all in into a Lie group approach, nice
- Richard Feynman's derivation of the Schrodinger equation:
- physics.stackexchange.com/questions/263990/feynmans-derivation-of-the-schrödinger-equation
- www.youtube.com/watch?v=xQ1d0M19LsM "Class Y. Feynman's Derivation of the Schrödinger Equation" by doctorphys (2020)
- www.youtube.com/watch?v=zC_gYfAqjZY&list=PL54DF0652B30D99A4&index=53 "I5. Derivation of the Schrödinger Equation" by doctorphys
There are unlisted articles, also show them or only show them.