What must I do with this Linear operator?

Question

Can anyone help help me with this problem

I want to show that the operator $\mathscr{L}=u \partial/\partial y$, is not an linear operator. This is what i got

$\mathscr{L}(au+bv)=u \partial/\partial y(au+bv) =u \partial/\partial y (au)+u \partial/\partial y(bv)=au\partial u/\partial y+bu\partial u/\partial y = a\mathscr{L}(u)+b\mathscr{L}(v)$

That seems linear to me, what did i do wrong?

Answer 1

your firts mistake is $$\mathscr{L}(au+bv)=u \partial/\partial y(au+bv)$$ it should be $\mathscr{L}(au+bv)=(au + bv) \partial/\partial y(au+bv)$

Answer 2

Take functions $u$ and $v$ defined by $u(x,y)=y$ and $v(x,y)=x$. Then: $$\mathscr{L}(u+v)=(u+v) (u+v)_y = u u_y +vu_y = u+v,$$ while $$\mathscr{L}(u)+\mathscr{L}(v) = uu_y +vv_y = u.$$