Is the differential forms perspective on $dx$ incompatible with the technique of implicit differentiation?
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Suppose $$x^2 + y^2 = 5^2.$$ We're trying to find $dy/dx$ at $(3,4).$ Applying $d$ to both sides: $$2x dx + 2y dy = 0$$ Or in other words: $$2x dx + 2y dy = 0dx + 0dy$$ Since the covectors $dx_p$ and $dy_p$ form a basis for the cotangent space at any $p in mathbb{R}^2$ , hence $2x = 0$ and $2y = 0.$ Hence $x = 0$ and $y = 0$ . Ergo $0^2 + 0^2 = 5^2$ , a contradiction. Question. Does this mean that the differential forms perspective on $dx$ is incompatible with the technique of implicit differentiation? If not, why not? If so, what definition of $dx$ can be used to avoid this issue?
calculus differential-geometry differential-topology differential-forms implicit-differentiation