Parametric Volume, volume analogue of parametric surface?

Question

Is it possible to have a parametric volume? Is it possible to have a parametric volume in $\mathbb{R}^3$, $V:\mathbb{R}^3\rightarrow\mathbb{R}^3$, where $V$ is some function $V(x,y,z) = A(x,y,z)\hat{i} + B(x,y,z)\hat{j} + C(x,y,z)\hat{k}$ ? In a parametric curve a curve is traced using the end point of the vector starting from the origin. In a parametric surface, there are two parameters which trace out the surface, so why not there be a parametric volume, where three parameters trace out the volume. I know it looks exactly like a vector field where every point in space gives you a unique vector, are the two perhaps somehow analogous, or is my entire thinking wrong and you cannot parametise a volume in this manor.

Answer 1

Sure. And if the mapping is regular in the sense that the Jacobian determinant of $A,B,C$ is non-zero, what you have is an alternative coordinate system in your region. You can also have a three-dimensional manifold in a space of higher dimension.