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Tony Stark
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What is hypersurface
What exactly is a hypersurface?
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In summary: 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.
- #2
AgentSmith
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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. 
- #3
mathman
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In general (mathematical) terms, consider an n-dimensional object. Its surface is called a hypersurface of n-1 dimensions.
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Nugatory
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mathman said:
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...
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mathman
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It depends on the definition of "object". If you allow things of infinite extent, like a half space, then the plane is a surface.Nugatory said:
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Nugatory
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mathman said:
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.
FAQ: What exactly is a hypersurface?
What is a hypersurface?
A hypersurface is a mathematical concept that refers to a surface in a space with more than three dimensions. In simpler terms, it is a higher-dimensional version of a two-dimensional surface, such as a sphere or a cone.
How is hypersurface meaning determined?
The meaning of a hypersurface is determined by its mathematical properties, such as its dimension, curvature, and topology. Additionally, the context in which the hypersurface is being studied can also affect its meaning, as it can have different interpretations in different fields of science.
What is the significance of studying hypersurfaces?
Studying hypersurfaces can provide insights into higher-dimensional spaces and help us understand complex systems and phenomena. It has applications in various fields, including physics, mathematics, and computer science.
Can hypersurfaces exist in the physical world?
While hypersurfaces may not exist in the physical world as we perceive it, they can be used to model and study physical phenomena. For example, Einstein's theory of general relativity uses hypersurfaces to describe the curvature of spacetime.
How are hypersurfaces visualized?
Hypersurfaces can be visualized through various methods, such as projections, cross-sections, and computer-generated images. The visualization technique used depends on the dimensionality of the hypersurface and the purpose of the study.