Integral of the Laplacian of a function that is constant on the sphere

Question

Take $B^n=\{x\in\mathbb{R}^n:|x|<1\}$.

Let $f(x)=(|x|^2-1)^{20}\sin(x_1+\cdots+x_n)$.

Find the integral $$\int\limits_{B^n}\Delta f(x)dx$$

I took $S^n=\{x\in\mathbb{R}^n:|x|=1\}$ and said that $\partial B^n=S^n$.

The function seems arbitrary to me, so I wanted to use the divergence theorem to say that

$$\int\limits_{B^n}\Delta f(x)dx=\int\limits_{S^n}\langle\nabla f,N\rangle dS=\int\limits_{S^n}0\ dS=0$$

Since $\Delta f=\text{div}(\nabla f)$ and $f(x)=0$ is constant when $x\in S^n$, so $\langle\nabla f,N\rangle=0$, too.

Is it true?

Edit: I understand what the comments are saying. But,

$$\nabla f_i=20(|x|^2-1)^{19}(2x_i)\sin(x_1+\cdots+x_n)+(|x|^2-1)^{20}\cos(x_1+\cdots+x_n)$$

So, $\nabla f_i=0$ in $S^n$, no?

Answer 1

$f(x)=0$ is constant when $x\in S^n$, so $\langle\nabla f,N\rangle=0$, too.

$f(x)=0$ being constant when $x\in S^n$ tells you that $\langle\nabla f,v\rangle=0$ for vectors $v$ that are tangent to the sphere - specifically not the normal vector.

Now given the edit, that makes more sense. I can't see any flaw. But the point is not that the function is constant on the sphere, rather that its derivatives vanish in all directions.