Compute the distributional derivative of $1_{B(0,1)}$ on $\mathbb{R}^{2}$：$\partial_{j}1_{B(0,1)}$

Question

Compute the distributional derivative of $1_{B(0,1)}$ on $\mathbb{R}^{2}$——$\partial_{j}1_{B(0,1)}$ where $j=1,2$ and $B(0,1)$ is a ball in $\mathbb{R}^{2}$

My own attempt is: Take $\varphi\in\mathcal{S}(\mathbb{R}^{2})$, we have $\langle \partial_{j}1_{B(0,1)},\varphi\rangle=-\langle 1_{B(0,1)},\partial_{x}\varphi\rangle=\int\int_{\mathbb{R}^{2}}1_{B(0,1)}\partial_{x}\varphi dxdy=\int_{-1}^{1}\int_{-\sqrt{1-y^{2}}}^{\sqrt{1-y^{2}}}\partial_{x}\varphi dxdy=\int_{-1}^{1}\varphi(\sqrt{1-y^{2}},y)-\varphi(-\sqrt{1-y^{2}},y)dy$,

What should I do next? Thanks! any suggestion are welcome!!!

Answer 1

$$\begin{align} \langle \partial_j1_{B(0,1)},\phi\rangle&=-\langle 1_{B(0,1)},\partial_j\phi\rangle\\\\ &=\iint_{B(0,1)} \hat x_j\cdot \nabla \phi\,dS\\\\ &=\hat x_j\cdot \iint_{B(0,1)} \nabla \phi\,dS\\\\ &=\hat x_j\cdot \oint_{\partial B(0,1)} \hat n \phi\,d\ell\\\\ &=\oint_{\partial B(0,1)} \hat n_j\phi\,d\ell\\\\ &=\int_0^{2\pi} (\hat x_j\cdot\hat r)\phi(1,\theta)\,d\theta\\\\ &=\int_0^{2\pi}\int_0^\infty \delta(r-1)(\hat x_j\cdot\hat r)\phi(r,\theta)\,r\,dr\,d\theta \end{align}$$

So, in distribution, we see that

$$\partial_j 1_{B(0,1)}=\delta(r-1)\hat x_j\cdot \hat r$$