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For a given smooth enough function $G(x,y)$ the equation $G(x,y)=c$ defines the smooth curve, the level curve. Suppose a point $\left(x^{*}, y^{*}\right)$ lies on this curve, i.e. is a solution of this equation. If for $\left(x^{*}, y^{*}\right)$ one has

$$ \frac{\partial G}{\partial y}\left(x^{*}, y^{*}\right) \neq 0 $$

and

$$ \frac{\partial G}{\partial x}\left(x^{*}, y^{*}\right) \neq 0 $$

then the locus of level curve $G(x, y)=c$ around $\left(x^{*}, y^{*}\right)$ can be thought of as the graph of a function $y=y(x)$ or $x=x(y)$, and the slope of this curve at $\left(x^{*}, y^{*}\right)$ with respect to $x$ axis is (by Implicit Function Theorem): $$ -\frac{\frac{\partial G}{\partial x}\left(x^{*}, y^{*}\right)}{\frac{\partial G}{\partial y}\left(x^{*}, y^{*}\right)} $$ and the slope of this curve with respect to $y$ axis is (by Implicit Function Theorem): $$ -\frac{\frac{\partial G}{\partial y}\left(x^{*}, y^{*}\right)}{\frac{\partial G}{\partial x}\left(x^{*}, y^{*}\right)} $$

But it seems we have two different slopes for one curve? How? I couldn't make sense of this. It will be a great help if anyone explain this with intuitive way.

WhyMeasureTheory
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    It's the slope of $y(x)$ vs. the slope of $x(y)$. These are of course reciprocals of each other, just like with lines. – Ian Oct 21 '21 at 13:08
  • Yes, I am aware of that @lan. But it seems both need to be equal for to be the slope at point $\left(x^{}, y^{}\right)$. Isn't that? Or my understanding make the statement vague? – WhyMeasureTheory Oct 21 '21 at 13:12
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    No, it's "change in y over change in x" in one and "change in x over change in y" in the other. You want the flexibility to think about things this way because of things like $G(x,y)=x^2+y^2=1$ at $(x,y)=(1,0)$ vs. at $(x,y)=(0,1)$. – Ian Oct 21 '21 at 13:13
  • I have post a question related to Implicit Function Theorem. Could you see this question, please? @lan – falamiw Oct 21 '21 at 13:33

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