Application of Gauss divergence theorem or Grren's formula in n dimension

Question

Let $B_\rho$ denotes the ball of radius $\rho$ and center at $0$ in $\mathbb{R}^n$ then prove the following: $$\int_{B_\rho}u_{x_i}dx=\int_{\partial B_\rho} u \cos\langle r,x_i\rangle dS $$.

I know the Gauss-Green's theorem $$\int_\Omega u_{x_i} dx=\int_{\partial \Omega} u \nu^i dS,$$ where $\nu=(\nu_1,\nu_2,\ldots, \nu_n)$ is the unit normal vector. But I am not able to conclude form there.