Consider a differentiable function $f:D\subset\mathbb R^m\mapsto \mathbb R^n$ with $m\le n$. I know if $m=1$ then $f(D)$ is called by "path", if $m=2$ then $f(D)$ is called by "surface" and if $m=3$ then $f(D)$ is called by "solid". I wish to know, what is the specific name of $f(D)$ if $m>3$. Is there a literature about this name? I have googled it but I didn't find yet.

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    $\begingroup$ We should be very happy that particular names don't exist, since this would often discourage abstraction. $\endgroup$ – Siminore Oct 30 '14 at 8:48
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    $\begingroup$ @Siminore: Yes, but mathematicians are always curious with the abstractions, aren't they? :D $\endgroup$ – Jlamprong Oct 30 '14 at 8:51
  • $\begingroup$ You might be interested in reading about manifolds. $\endgroup$ – littleO Nov 3 '14 at 9:55
  • $\begingroup$ @littleO: Yes, it is related to smooth manifold, isn't it? $\endgroup$ – Jlamprong Nov 3 '14 at 10:37
  • $\begingroup$ Under certain assumptions on $f$, the set $f(D)$ will be a smooth manifold, which might be the generalization of the idea of a "surface" that you are looking for. $\endgroup$ – littleO Nov 3 '14 at 11:02

Actually, more is required. The image of $f$ is $m$-dimensional if the rank of $f$ is $m$.

About the question, the only other name frequently used is hypersurface when the dimension of the image is $n-1$.

  • $\begingroup$ Isn't there any name if the dimension of the image is $m$? $\endgroup$ – Jlamprong Nov 3 '14 at 10:39
  • $\begingroup$ @Jlamprong, I don't know any beyond the obvious "$m$-dimensional". $\endgroup$ – Martín-Blas Pérez Pinilla Nov 3 '14 at 10:59

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