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Happiness
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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.
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?
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?