Partial Derivative of Multivariate Indicator Function in the Distributional Sense Define $f:\mathbb{R}^2\to\mathbb{R}$ by $\left(x,y\right)\mapsto1$ if $x^2+y^2\leq1$ and $\left(x,y\right)\mapsto0$ otherwise.
Is there a definition of $\partial f/\partial x$ in the distributional sense? If so, what is it? References are welcome.
I know that $\partial f/\partial x=0$ on $\left\{\left(x,y\right)\in\mathbb{R}^2:x^2+y^2\neq1\right\}$, and my intuition tells me that $\partial f/\partial x$ behaves like the Dirac delta function on $\left\{\left(x,y\right)\in\mathbb{R}^2:x^2+y^2=1\right\}$, but I do not know how to express this rigorously.
 A: let $\vec x = (x_1,\dots,x_n)$, and let $\vec\phi:\mathbb R^n \to \mathbb R^n $ be a vector valued test function, extend the pairing between smooth functions of compact support $\langle \vec f,\vec g\rangle := \int_{\mathbb R^n} \vec f\cdot\vec g$ to distributions analagously to the scalar case. By extending Gauss's theorem, the natural generalisation of integration by parts to higher dimensions, we have the natural generalisation of the distributional derivative,
$$\langle \nabla f, \vec\phi\rangle :=  -\langle f,\nabla\cdot \vec\phi \rangle = -\int_{|\vec x|<1} \nabla \cdot \vec\phi(\vec x)\ d\vec x = -\int_{|\vec x|=1} \vec\phi\cdot \vec n \ dl$$
so $\nabla f$ is a vector valued distribution supported on $\mathbb S^1 = \{|\vec x|=1\}$ which points in the direction of the outward normal from $\mathbb S^1$. If you wanted to, you could define for any test function $\varphi : \mathbb R^n \to \mathbb R$, 
$$\langle \delta_{\mathbb S^1},\varphi\rangle := \int_{|\vec x|=1} \varphi \ dl,$$
then (somewhat abusively since $\vec n$ is only defined on $\mathbb S^1$), $$\nabla f = -\vec n \delta_{\mathbb S^1}$$
The components $\partial_{i} f$ are related as follows. If $\varphi : \mathbb R^n \to \mathbb R$ is a test function, set $\vec\phi = \varphi \vec e_i$, where $\vec e_i$ is the $i$th standard basis vector of $\vec R^n$. Then
$$\langle \partial_i f, \varphi\rangle := - \langle f , \partial_i \varphi\rangle = - \langle f,\nabla \cdot \vec\phi\rangle = \langle \nabla f,\vec\phi\rangle = \langle -\vec n \delta_{\mathbb S^1} , \vec\phi\rangle = \langle -n_i\delta_{\mathbb S^1}, \varphi\rangle$$
i.e. $$ \frac{\partial f}{\partial x_i} = -n_i\delta_{\mathbb S^1}$$
in the sense of distributions.
I don't have any references for the above computation; it is not difficult, but I have never seen it before. It would be interesting to have a theory of generalised $n$-forms that would allow both the above and the usual computations with generalised functions, but I don't know of it. So I would also greatly appreciate some references.
