Take $\mathbb R^2$ as example, it is known that $\delta_{x_1=0}(\phi)=\int_{\mathbb R}\phi(0,x_2)dx_2$. What about $\delta_{x_2=x_1^2}$?
At the beginning I thought it should be integrated by arc length i.e. $\delta_{x_2=x_1^2}(\phi)=\int_{x_2=x_1^2}\phi(x_1,x_2)ds=\int_{\mathbb R}\phi(t,t^2)\sqrt{4t^2+1}dt$. But this definition does not preserve linear scaling.
Is there any best definition to characterize these Dirac function on a curve?