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For a function $F(x,y,z)$ with $(a,b,c)$ on the level surface $F(x,y,z)=k$, where $F(x,y,z)=k$ defines $z$ implicitly as a function of $x$ and $y$. Using the chain rule, assuming $F_z(a,b,c)\neq0$ show that at the point $(a,b,c)$,

$\frac{\partial z}{\partial x}=g_x(a,b)=-\frac{F_x(a,b,c)}{F_z (a,b,c)}$

I start by using the chain rule to differentiate w.r.t $x$

$F_x=\frac{\partial F}{\partial x} \frac{dx}{dx}+\frac{\partial F}{\partial y} \frac{dy}{dx} + \frac{\partial F}{\partial z} \frac{dz}{dx}$

In the teacher's hint, he stated that $\frac{dy}{dx}$ is equal to zero, which using this I can solve the question, I don't however understand why $\frac{dy}{dx}=0$. I was wondering if someone could explain this.

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    $\begingroup$ Z is a function of x and y but x and y are independent variables and thus not functions of each other $\endgroup$ – Triatticus Feb 4 '16 at 7:48
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Get the total differential of F(x,y,z)=k

$$dF(x,y,z)=\frac{\partial F}{\partial x}dx+\frac{\partial F}{\partial y}dy+\frac{\partial F}{\partial z}dz=0$$ Since z is implicitly defined by x and y $$dz=\frac{\partial z}{\partial x}dx+\frac{\partial z}{\partial y}dy$$ Now replacing dz to dF $$dF(x,y,z)=\frac{\partial F}{\partial x}dx+\frac{\partial F}{\partial y}dy+\frac{\partial F}{\partial z}(\frac{\partial z}{\partial x}dx+\frac{\partial z}{\partial y}dy)=0$$ Rearranging the terms $$dF(x,y,z)=\bigg(\frac{\partial F}{\partial x}+\frac{\partial F}{\partial z}\frac{\partial z}{\partial x}\bigg)dx+\bigg(\frac{\partial F}{\partial y}+\frac{\partial F}{\partial z}\frac{\partial z}{\partial y}\bigg)dy=0$$ To make this equation hold we need to satisfy below conditions $$\frac{\partial F}{\partial x}+\frac{\partial F}{\partial z}\frac{\partial z}{\partial x}=0 \Rightarrow \frac{\partial z}{\partial x}=-\frac{\frac{\partial F}{\partial x}}{\frac{\partial F}{\partial z}}$$ $$\frac{\partial F}{\partial y}+\frac{\partial F}{\partial z}\frac{\partial z}{\partial y}=0 \Rightarrow \frac{\partial z}{\partial y}=-\frac{\frac{\partial F}{\partial y}}{\frac{\partial F}{\partial z}}$$ The first condition is what you are looking for.

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