I want to derive the Euler-Lagrange equations (ELE) from an action $$S=\int\int\mathcal{L}(x, y, z, \partial_y x, \partial_zx) dydz$$ where $x$ depends on two variables $x=x(y,z)$, imposing $\delta S=0$, but I find some extra terms that I don't think should be there.
I proceeded as follows:
$$0=\delta S=\delta\left(\int\int\mathcal{L}dydz \right)=\int\int\delta(\mathcal{L})dydz=\int\int\left(\frac{\partial\mathcal{L}}{\partial x}\delta x+\frac{\partial\mathcal{L}}{\partial y}\delta y+\frac{\partial\mathcal{L}}{\partial z}\delta z+\frac{\partial\mathcal{L}}{\partial \partial_yx}\delta \partial_yx+\frac{\partial\mathcal{L}}{\partial \partial_zx}\delta \partial_zx\right)dydz=$$
$$=\int\int\left(\frac{\partial\mathcal{L}}{\partial x}+\frac{\partial\mathcal{L}}{\partial y}\frac{\partial y}{\partial x}+\frac{\partial\mathcal{L}}{\partial z}\frac{\partial z}{\partial x}-\partial_y \frac{\partial\mathcal{L}}{\partial \partial_yx}-\partial_z \frac{\partial\mathcal{L}}{\partial \partial_zx}\right)\delta xdydz+\text{boundary conditions}$$
$$\Rightarrow \frac{\partial\mathcal{L}}{\partial x}+\frac{\partial\mathcal{L}}{\partial y}\frac{\partial y}{\partial x}+\frac{\partial\mathcal{L}}{\partial z}\frac{\partial z}{\partial x}=\partial_y \frac{\partial\mathcal{L}}{\partial \partial_yx}+\partial_z \frac{\partial\mathcal{L}}{\partial \partial_zx}$$
if the boundary conditions are met.
Is this formula the correct ELE in 2D? How should I treat the $\partial y / \partial x$, $\partial z / \partial x$?
Does this change if the extrema of the integrals are functions of the variables? For example, if my action is $$S=\int_0^1 dz\int_{\hat{y}(z)}^1dy\mathcal{L}(x, y, z, \partial_y x, \partial_zx)$$