Near field approximation to Kirchhoff's diffraction formula, i.e. when the plane of observation is near the object diffracting.
Completeness: math.stackexchange.com/questions/2192665/is-this-set-of-bessel-functions-a-basis-for-all-c10-a-functions TODO
This is the Bessel function analogue to Fourier basis is complete for .
Existence and uniqueness of solutions of partial differential equations by 
Ciro Santilli 37 Updated 2025-07-16
Ciro Santilli 37 Updated 2025-07-16Unlike for ordinary differential equations which have the Picard–Lindelöf theorem, the existence and uniqueness of solution is not well solved for PDEs.
For example, Navier-Stokes existence and smoothness was one of the Millennium Prize Problems.
In many important applications, what you have to solve is not just a single partial differential equation, but multiple partial differential equations coupled to each other. This is the case for many key PDEs including:
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.
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
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