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I am confused about antiderivatives of multivariable functions, specifically $\delta(ct-|x|)$ and $\delta(t-|x/c|)$.

Here $\delta(.)$ is the Dirac delta function (distribution)and $x$ and $t$ are variables and $c$ is a constant $>0$.

MY QUESTION IS: What variable(s) are the antiderivative integrals of $\delta(ct-|x|)$ or $\delta(t-|x/c|)$ with respect to, are there separate antiderivatives for each variable (what do they look like), is there such a thing as a 'total antiderivative'?

I know these antiderivatives lead to Heaviside step functions.

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    $\begingroup$ Usually the antiderivative is taken of $\delta(\cdot)$, but it's generally impossible to say without a context. $\endgroup$ – md2perpe Nov 7 at 7:06
  • $\begingroup$ I edited my question. Hope that clarifies things. $\endgroup$ – user45664 Nov 7 at 17:04
  • $\begingroup$ Still: Your question can not be answered without context. One can take the antiderivative w.r.t. all of $ct-|x|$ (or $t-|x/c|$), w.r.t. $t$, or try to find some w.r.t. $x$. $\endgroup$ – md2perpe Nov 7 at 17:42
  • $\begingroup$ So I think you answered most of my question--it was about those alternatives. Can you write that up with the integrals as an answer? $\endgroup$ – user45664 Nov 11 at 17:50
  • $\begingroup$ Are you familiar with the "jump formula" for the derivative of a piece-wise $C^1$ function? Consider the indicator function defined as $f(x,t) = 1$ for $t\ge |x|$ and $f(x,t)=0$ for $t < |x|$. Can you find its derivative w.r.t. $x$ for a given $t\ne0$? $\endgroup$ – TZakrevskiy Nov 12 at 14:46

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