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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.

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up vote 5 down vote accepted

Yes, but for rather silly reasons: when you consider the partial derivative with respect to $x$, you are taking $y$ and $z$ to be fixed. Being fixed, they are constant relative to a change in $x$, hence $$\frac{\partial y}{\partial x} = \frac{\partial z}{\partial x} = 0.$$ That is, the values of $y$ and of $z$ do not depend on $x$, so their partial derivatives are zero. The fact that $F_y$ and $F_z$ are nonzero is irrelevant.

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