Lie group-Lie algebra correspondence by 
Ciro Santilli 37 Updated 2025-05-07 +Created 1970-01-01
Ciro Santilli 37 Updated 2025-05-07 +Created 1970-01-01Bibliography:
An Introduction to Tensors and Group Theory for Physicists by Nadir Jeevanjee (2011) shows that this is a tensor that represents the volume of a parallelepiped.
It takes as input three vectors, and outputs one real number, the volume. And it is linear on each vector. This perfectly satisfied the definition of a tensor of order (3,0).
A measurable function defined on a closed interval is square integrable (and therefore in ) if and only if Fourier series converges in norm the function:
An LC circuit is analogous to a spring-mass system by Ciro Santilli 37 Updated 2025-05-07 +Created 1970-01-01
Ciro Santilli 37 Updated 2025-05-07 +Created 1970-01-01Both are harmonic oscillators.
In the LC circuit:
- the current current may be seen as the velocity and containing the kinetic energy
- the charge stored in the capacitor as the potential energy
You can kickstart motion in either of those systems in two ways:
You can learn more from older students than from faculty by Ciro Santilli 37 Updated 2025-05-07 +Created 1970-01-01
Ciro Santilli 37 Updated 2025-05-07 +Created 1970-01-01Wikipedia mentions quoting his Nobel Prize biography:
TODO confirm: does the solution of the heat equation always converge to the solution of the Laplace equation as time tends to infinity?
In one dimension, the Laplace equation is boring as it is just a straight line since the second derivative must be 0. That also matches our intuition of the limit solution of the heat equation.
Non-invertible are excluded "because" otherwise it would not form a group (every element must have an inverse). This is therefore the largest possible group under matrix multiplication, other matrix multiplication groups being subgroups of it.
TODO motivation. Motivation. Motivation. Motivation. The definitin with quotient group is easy to understand.
Our notation: , called "dihedral group of degree n", means the dihedral group of the regular polygon with sides, and therefore has order (all rotations + flips), called the "dihedral group of order 2n".
Ames Hydroelectric Generating Plant by Ciro Santilli 37 Updated 2025-05-07 +Created 1970-01-01
Ciro Santilli 37 Updated 2025-05-07 +Created 1970-01-01 Inward Bound by Abraham Pais (1988) by Ciro Santilli 37 Updated 2025-05-07 +Created 1970-01-01
Ciro Santilli 37 Updated 2025-05-07 +Created 1970-01-01The book unfortunately does not cover the history of quantum mechanics very, the author specifically says that this will not be covered, the focus is more on particles/forces. But there are still some mentions.
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