Consider a surface defined by the equation ##g(x, y, z)=0##. The intersection between this surface and the plane ##z=c## produces a curve that can be plotted on an x-y plane. Find the gradient of this curve.(adsbygoogle = window.adsbygoogle || []).push({});

By chain rule,

##\frac{\partial y}{\partial x}=\frac{\partial y}{\partial g}\frac{\partial g}{\partial x}##

Using the reciprocity relation ##\frac{\partial y}{\partial g}=\Big(\frac{\partial g}{\partial y}\Big)^{-1}##, we have

##\frac{\partial y}{\partial x}=\frac{(\frac{\partial g}{\partial x})}{(\frac{\partial g}{\partial y})}##

This differs from the correct answer by a negative sign. What's wrong with this method?

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# I Derive the formula for gradient using chain rule

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