Notation For Delta Distribution

Question

I've encountered two different notations for the delta distribution: $$\delta(x,y)$$ and $$\delta(x-y)$$ What is the difference between these two notations. Do they depend on context or should I prefer one over the other?

Answer 1

The first notation is more convenient when working in high dimensions, since it naturally generalizes to $n$ dimensions: e.g., $\delta (x,y,z,w, ...)$. You can do this with the second notation $\delta(x) \delta(y), ...$

Certain software (e.g., Mathematica) supports both usages.

Answer 2

The first describes a delta function acting at the origin.

The second describes a delta function acting along the line $x=y$: