Line integral of multi dimensional Dirac Delta

Question

How do we compute the line integral of a Dirac Delta? e.g. $\int_a^b \delta(x-x(t),y-y(t))dt$ ? Consequently, is $\int_{-\infty}^{\infty}\delta(x-t)\delta(y-t)=\delta(x-y)$ or $\delta((x-y)/\sqrt2)$? I'm more interested in the general case though rather than this specific example.

Answer 1

We have $\int_a^b \delta(x-x(t)),y-y(t))dt = \sum_{\{t: x(t)=x, y(t)=y\}}1$

So $\int_{-\infty}^\infty \delta(x-t)\delta(y-t) = \delta(x-y)$