# nice name for the image of multivariable function

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.

• We should be very happy that particular names don't exist, since this would often discourage abstraction. – Siminore Oct 30 '14 at 8:48
• @Siminore: Yes, but mathematicians are always curious with the abstractions, aren't they? :D – Jlamprong Oct 30 '14 at 8:51
• You might be interested in reading about manifolds. – littleO Nov 3 '14 at 9:55
• @littleO: Yes, it is related to smooth manifold, isn't it? – Jlamprong Nov 3 '14 at 10:37
• 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. – 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 $n-1$.
• Isn't there any name if the dimension of the image is $m$? – Jlamprong Nov 3 '14 at 10:39
• @Jlamprong, I don't know any beyond the obvious "$m$-dimensional". – Martín-Blas Pérez Pinilla Nov 3 '14 at 10:59