I ask myself if following mappings with cartesian product involved are true ($f$ is continuous):
$\begin{align} x\in[a,b]\subset \mathbb{R}\rightarrow \left(\begin{array}{cc}f_1(x)\\f_2(x)\\f_3(x)\end{array}\right) \end{align}$ describes a line in $\mathbb{R}^3$
$\begin{align} (x,y)\in[a,b]\times[c,d]\subset \mathbb{R^2}\rightarrow \left(\begin{array}{cc}f_1(x)\\f_2(x)\\f_3(x)\end{array}\right) \end{align}$ describes a plane in $\mathbb{R}^3$
$\begin{align} (x,y,z)\in[a,b]\times[c,d]\times[e,f]\subset \mathbb{R^3}\rightarrow \left(\begin{array}{cc}f_1(x)\\f_2(x)\\f_3(x)\end{array}\right) \end{align}$ describes a volume in $\mathbb{R}^3$
These lines/planes/volumes would move according to the vector field $f(\textbf{x})$.