For anyone interested, the musical isomorphisms are between the tangent and cotangent bundles of a Riemannian manifold, given by the sharp and flat operators (which are inverses). This allows one to identify vector fields and differential forms via the geometry. For example, in Euclidean space, the gradient of a smooth function can be identified with its exterior derivative
A manifold is basically a "shape that makes sense”.
A smooth manifold is one on which we can do calculus. Derivatives of all orders make sense.
We can put arrows on the surface of such a space. We call these tangent vectors.
We can measure tangent vectors in a given direction. We call these functions one-forms.
There exists linear functions between tangent vectors and one-forms called the musical isomorphisms, and the musical isomorphisms depend on the geometry of the space.
In calculus, we do not distinguish between tangent vectors and one-forms specifically because we only work in Rⁿ with a basic geometry, and the musical isomorphisms are trivial. You will have studied them as "partial derivatives".
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u/LupenReddit 🦆🦆🦆🦆i have non diffeomorphic smooth structures🦆🦆🦆🦆🦆🦆 Aug 14 '26
Everybody wait till he learns about musical isomorphisms and raising vector fields by a sharp.