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

Nabla symbol ( $\nabla$ )

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.

Del

Oh, and if it weren't enough, mathematicians have a separate name for the damned nabla symbol : "del" instead of "nabla".

TODO why is it called "Del"? Is is because it is an inverted uppercase delta?

Divergence ( $\nabla \cdot$ , $d i v ()$ )

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:

the input has to be the 6 flows across each face, therefore 3 derivatives
the output is the variation of the quantity of fluid, and therefore a scalar

Gradient ( $\nabla$ )

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:

take a scalar field as input. Otherwise, how do you decide which vector is larger than the other?
output a vector field that contains the direction in which the scalar increases fastest locally at each point. This has to give out vectors, since we are talking about directions