# Simple question on the antiderivative of a two variable function (or distribution)?

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.

• Usually the antiderivative is taken of $\delta(\cdot)$, but it's generally impossible to say without a context. – md2perpe Nov 7 at 7:06
• I edited my question. Hope that clarifies things. – user45664 Nov 7 at 17:04
• 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. $x$. – md2perpe Nov 7 at 17:42
• 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? – user45664 Nov 11 at 17:50
• 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 $t\ne0$? – TZakrevskiy Nov 12 at 14:46