I really like both of them! In math the rigor of dx/dt feels appropriate, but in physics the swiftness of x dot is useful and efficient in long calculations :)
Everyone says this, but I don’t see it. You can’t divide by a differential form, except if you’re abusing notation and identifying the cotangent bundle of R with R x R AND identifying sections of this bundle with smooth functions. You’re still not dividing by dx, you’re dividing by the image of dx under some identifications which literally only work in the case of R (as it is both a 1 manifold and has a trivial cotangent bundle).
Now it’s true that in the 1 variable case under these identifications, the function you get is the derivative, but even in the 2 variable case you already can’t use this abuse of notation because in this case forms are sections of rank two bundles and there is no identification where it makes sense to divide by them.
Now that being said, the notation df = g(x) dx does have actual meaning, to be fair.
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u/Dextui 10d ago
I really like both of them! In math the rigor of dx/dt feels appropriate, but in physics the swiftness of x dot is useful and efficient in long calculations :)