I am curious to know some theorems usually taught in advanced math courses which are considered 'gen

Jovany Clayton 2022-07-03 Answered
I am curious to know some theorems usually taught in advanced math courses which are considered 'generalizations' of theorems you learn in early university or late high school (or even late university).
For example, I know that Stokes's theorem is a generalization of the divergence theorem, the fundamental theorem of calculus and Green's theorem, among I'm sure many other notions.
I've read that pure mathematics is concerned mostly with the concept of 'generalization' and I am wondering which theorems/ideas/concepts, like Stokes's theorem, are currently celebrated 'generalizations' by mathematicians.
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Answers (2)

Nirdaciw3
Answered 2022-07-04 Author has 20 answers
The Yoneda lemma of category theory says that every category can be understood as a category whose objects are sets and whose morphisms are functions.
A group can be construed as a special case of a category: it is a category with only one object, whose morphisms are all invertible. The special case of the Yoneda lemma for such a category says that the (single) object can be understood as a set S and the morphisms (which are the group elements) can be understood as invertible functions from S to itself.
This is exactly Cayley's theorem of group theory; it says that any group can be understood as a group of permutations of the elements of some set.
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Cierra Castillo
Answered 2022-07-05 Author has 6 answers
In spherical geometry, the area of a triangle on a unit sphere with angles A , B , C is
A + B + C π ,
a result that dates to maybe the 17th century.
The Gauss-Bonnet Theorem generalizes this to any compact surface (2-dimensional Riemannian manifold with corners) ( M , g ): If ( M , g ) has Gaussian curvature K and its boundary M has geodesic curvature k g , we have
M K d s + M k g = 2 π χ ( M ) ,
where χ ( M ) is the Euler characteristic. (Note the in interpreting the boundary integral, we must include the sum of the turning angles at each of the corners.)
I suppose this also generalizes the classic high school result that the sum of the angles of a plane triangle is π, as well as the formula for the circumference of a unit circle.
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