Ciro Santilli @cirosantilli 35

A vector field with a bilinear map into itself, which we can also call a "vector product".

Note that the vector product does not have to be neither associative nor commutative.

Examples: en.wikipedia.org/w/index.php?title=Algebra_over_a_field&oldid=1035146107#Motivating_examples

First we write a vector field as:Note how we are denoting each component of $F$ as $F^i$ with a raised index.

Then, the divergence can be written in Einstein notation as:

It is common to just omit the variables of the function, so we tend to just say:or equivalently when referring just to the operator:

In the context of Maxwell's equations, it is vector field that is one of the inputs of the equation.

Section "Maxwell's equations with pointlike particles" asks if the theory would work for pointlike particles in order to predict the evolution of this field as part of the equations themselves rather than as an external element.

Measured in amperes in the International System of Units.

Basically, a "representation" means associating each group element as an invertible matrices, i.e. a matrix in (possibly some subset of) $GL(n)$, that has the same properties as the group.

Or in other words, associating to the more abstract notion of a group more concrete objects with which we are familiar (e.g. a matrix).

Each such matrix then represents one specific element of the group.

This is basically what everyone does (or should do!) when starting to study Lie groups: we start looking at matrix Lie groups, which are very concrete.

Or more precisely, mapping each group element to a linear map over some vector field $V$ (which can be represented by a matrix infinite dimension), in a way that respects the group operations:

As shown at Physics from Symmetry by Jakob Schwichtenberg (2015)

Motivation:

Bibliography: