Cliff Notes

Condensed cheat sheet/mnemonics of stuff I forget

Inner Product => Angle Norm => Length Metric => Distance Measure => Size LpL_{p} norm => Max() component L0L_{0} norm => Counting norm

Gradient => derivative Jacobian => how a differential patch area is skewed under a (linear only?) transformation Divergence => sink vs source aka volume density of outward flux Laplacian => average of neighborhood

Manifold - fancy name of a curved space Reimannian Manifolds Lie Group - a group that is a manifold (aka a group where group multiplication is smooth aka infinitely differentiable)

  • Group requirementrs: closed under multiplication, commutative, identity function, inverse Lie Algebra: tangent space of Lie group

Banach Spaces Hilbert Spaces Sobolev Space

Functionals => functions that take functions as inputs (derivative/integral operators)

A manifold is a fancy name of curved space •A matrix group is a curved space with a group structure:it is called a Lie group. •A tangent space is a linear approximation of a curved space •A tangent space of Lie group is called Lie algebra. •Dual number is convenient for computation of Lie algebra