A hypersurface is a generalization of an ordinary two-dimensional surface embedded in three-dimensional space to an 
-dimensional surface embedded in 
-dimensional space. Locally, a smooth hypersurface can be written
as the regular level set
 |
where the gradient of 
does not vanish, and so it has codimension
one. For instance, the 
-dimensional
hypersphere 
in 
corresponds to the equation 
.
