Lets $F(x, y, z):\mathbb{R}^{3}\to\mathbb{R}^{1}$, and lets $F_y\neq 0$ and $F_z\neq 0$ in some neighborhood $V$ of $(x_0, y_0, z_0)$.
Am I right that: $\frac{\partial F}{\partial x}=\frac{\partial F}{\partial x}\frac{\partial x}{\partial x}+\frac{\partial F}{\partial y}\frac{\partial y}{\partial x}+\frac{\partial F}{\partial z}\frac{\partial z}{\partial x}$ and hence $\frac{\partial F}{\partial y}\frac{\partial y}{\partial x}+\frac{\partial F}{\partial z}\frac{\partial z}{\partial x}=0$ in $V\ni(x_0, y_0, z_0).$
Thanks.