I am reading Einstein Manifolds (Besse). At the end of Section 5F, the author states (paraphrasing slightly):
"One may have been confused that the Ricci equation $\text{Ric}(g) = r$ and the Einstein equation $\text{Ric}(g) = \lambda g$ are elliptic in certain coordinates, but not in others (whereas, "the definition of "elliptic" is invariant under changes of coordinates").
This is resolved by observing that in changing coordinates for these equations, one not only changes the independent variables, but also one is making a change in the dependent variables as well, since the metric $g$ is a tensor field and hence is transformed, too. The statement that "ellipticity is invariant" refers only to changes of independent variables."
I only grasp these two paragraphs on a hazy, vague level. To gain a better understanding, I would like to see a simpler example that explicitly illustrates how "changing the independent variables" versus "changing the dependent variables" affects (or doesn't affect) the ellipticity of the equation.
I would be especially interested in seeing this illustrated with, say, the Laplace equation $\Delta u = 0$ for functions $u \colon U \subset \mathbb{R}^n \to \mathbb{R}$, or the harmonic map system $\Delta \mathbf{u} = \mathbf{0}$ for functions $\mathbf{u} \colon U \subset \mathbb{R}^n \to \mathbb{R}^k$, or the Cauchy-Riemann system $u_x = v_y$, $u_y = -v_x$ for functions $(u,v) \colon \mathbb{R}^2 \to \mathbb{R}^2$.
This question is closely related to this very nice one by juan rojo and also to one that I asked a while ago.
