$$\frac{\partial^{2} u}{\partial s^{2}}=\frac{\partial}{\partial s}\left(xu_{x}+yu_{y}\right)=xu_{x}+xu_{xs}+yu_{y}+yu_{ys},$$
And:
$$\frac{\partial^{2} u}{\partial t^{2}}=\frac{\partial}{\partial t}\left(-yu_{x}+xu_{y}\right)=-xu_{x}-yu_{xt}-yu_{y}+xu_{yt}$$
Like I really don't understand. It looks simple, but I don't understand the rule. There's nothing in my textbook that explains this. I think it is somehow related to the chain rule, but I don't really see how it could be tied to that.
$$u = f(x,y)$$
