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!
Oh, and if it weren't enough, mathematicians have a separate name for the damned nabla symbol : "del" instead of "nabla".
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
Mnemonic: the gradient shows the direction in which the function increases fastest.
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
Can be denoted either by:Our default symbol is going to be:
- the upper case Greek letter delta
- nabla symbol squared
The laplace operator for Minkowski space.
Can be nicely written with Einstein notation as shown at: Section "d'Alembert operator in Einstein notation".
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