Ciro Santilli @cirosantilli  40

Directly modelled by group.

For continuous symmetries, see: Lie group.

A Ring can be seen as a generalization of a field where:

Addition however has to be commutative and have inverses, i.e. it is an Abelian group.

The simplest example of a ring which is not a full fledged field and with commutative multiplication are the integers. Notably, no inverses exist except for the identity itself and -1. E.g. the inverse of 2 would be 1/2 which is not in the set. More specifically, the integers are a commutative ring.

A polynomial ring is another example with the same properties as the integers.

The simplest non-commutative, non-division is is the set of all 2x2 matrices of real numbers:Note that $GL(n)$ is not a ring because you can by addition reach the zero matrix.

Points in the direction in which a wind spinner spins fastest.

As if Greek letters weren't enough, physicists and mathematicians also like to make up tons of symbols, some of which look like the could actually be Greek letters!

Nabla is one of those: it was completely made up in modern times, and just happens to look like an inverted upper case delta to make things even more confusing!

Nabla means "harp" in Greek, which looks like the symbol.

Takes a vector field as input and produces a scalar field.

Mnemonic: it gives out the amount of fluid that is going in or out of a given volume per unit of time.

Therefore, if you take a cubic volume:

Takes a scalar field as input and produces a vector field.

Mnemonic: the gradient shows the direction in which the function increases fastest.

Think of a color gradient going from white to black from left to right.

Therefore, it has to: