Proof by Notation (yes, it's actually true)pic.twitter.com/gTFvdfTzIw
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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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I think that physicists should always use scare-quotes when they talk about "proving" stuff :p. The proof assumes that the inverse of a differentiable function is differentiable which is part of what you are trying to prove!
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the statement i'm making is when the derivative of the inverse of f and f exist when you get dy/dx dx/dy = 1
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wait, this is very much not called that: the Inverse Function Theorem says that if the derivative is nonzero at a point, then the function is invertible on some neighborhood around that point!
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As a physics major, this makes sense, but I've been told that treating differentials like standard algabraic variables will give mathematicians conniptions.
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There are so many things wrong with this proof: You can’t assume an inverse function exists without first assuming bijectivity or demonstrating it d/dx is a linear operator so your algebra is invalid
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You should do one on fubinis theorem for us physicists.
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You first need to prove that, assuming f is differentiable at c∈R, then f^-1 is differentiable at d=f(c). This proof just assumes it.
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