Let $p:\Bbb R^n\times (0,\infty)\to\Bbb R,\:(x,y)\mapsto y/(|x|^2+y^2)$. Calculate $\Delta p$.
Here $\Delta$ is the laplacian and $|x|^2=(x|x)$ is the squared euclidean norm. I need to check if the exercise is correct, can some one take a look at it? We have that
$$\partial_x p(x,y)v=-\frac{2(x|v)y}{(|x|^2+y^2)^2},\quad\partial_y p(x,y)=\frac1{|x|^2+y^2}-\frac{2y^2}{(|x|^2+y^2)^2}$$
where I used the fact that $\partial |x|^2h=2(x|h)$. Then
$$\partial^2_x p(x,y)[v,h]=-\frac{2y(v|h)}{(|x|^2+y^2)^2}+\frac{8y(x|v)(x|h)}{(|x|^2+y^2)^3}$$
$$\partial^2_y p(x,y)=-\frac{6y}{(|x|^2+y^2)^2}+\frac{8y^3}{(|x|^2+y^2)^3}$$
But now it is not clear how to write the laplacian, maybe
$$\Delta p(x,y)[v,h]\overset{?}{=}8y\frac{y^2+(x|v)(x|h)}{(|x|^2+y^2)^3}-2y\frac{3+(v|h)}{(|x|^2+y^2)^2}$$
Ok, I solved it. The definition of laplacian in the book is
$$\Delta:C^2(X,\Bbb R)\to C(X,\Bbb R),\quad u\mapsto \sum_{k=1}^m\partial^2_{x_k}u$$
for $X\in\Bbb R^m$.
Thus applied this to the above question it is not enough to differentiate respect to each independent variable, if not for each dimension of the variables of the function, that is, for every of it coordinates because
$$\Bbb R^n\times (0,\infty)\subset\Bbb R^{n+1}$$
Then applying this to the above is easy to see that $\Delta p=0$.