What is hypersurface
Its a surface of something with more than three dimensions. Think of a circle, then a sphere, then a hypersphere. A hypersphere has (at least) one more dimension than a sphere, even though we can't visualize it. The hypersphere's surface is called a hypersurface. You can have a hypercube with a hypersurface. All of these are mathematical objects.
In general (mathematical) terms, consider an n-dimensional object. Its surface is called a hypersurface of n-1 dimensions.
That definition is intuitive but a bit too limiting. For example, the plane ##x=17## is a two-dimensional hypersurface in three-dimensional Euclidean space, but it not the surface of any three-dimensional object.
Mathematically, an n-dimensional hypersurface is an n-dimensional submanifold of an (n+1)-dimensional manifold. Examples include mathman's two-dimensional surface of a three-dimensional sphere; the three-dimensional surfaces of simultaneity (constant t coordinate in a given frame) in four-dimensional space-time; just about any two-dimensional surface, whether curved or flat, in three-dimensional Euclidean space....
It depends on the definition of "object". If you allow things of infinite extent, like a half space, then the plane is a surface.
Indeed it is.
It's conversations like this one that explain why we need the rigorous definitions as well as the intuitive example-based explanations.
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