This result is known as "Inverse Function Theorem".
Here's a proof using the chain rulepic.twitter.com/KiqhlW78Cr
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The proper generalization of the Inverse Function Theorem to higher dimensions applies to the 𝐽𝑎𝑐𝑜𝑏𝑖𝑎𝑛 𝑚𝑎𝑡𝑟𝑖𝑥 (defined below)
[ ]⁻¹ denotes the 𝑚𝑎𝑡𝑟𝑖𝑥 inversepic.twitter.com/wyvtEcYOfQ
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Behold the surreal power of Leibniz notation
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Fun fact: this is not true in general for partial derivatives. Example: ∂r/∂x = x/r = cosθ, but 1/(∂x/∂r) = 1/cosθ.
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I think the proper generalization is with the jacobian where ∂(x,y)/∂(u,v) = [∂(u,v)/∂(x,y)]^{-1} where ^{-1} denotes the matrix inverse
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Except when dx/dy=0?
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If dx/dy=0, then dy/dx must diverge to infinity, so it is still valid in a sense
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not necessarily for differentials ;)
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