The Taylor expansion of a vector field $f(x)$ to the order of one is $$f(x)=f(x_0)+Jf(x_0)\cdot\Delta x+o(\Delta x)$$ where $Jf$ is Jacobian of the vector field and $\Delta x=x-x_0$.

Suppose we decompose it into symmetric and skew-symmetric part such that $$S=\frac{Jf+(Jf)^T}{2}$$ $$A=\frac{Jf-(Jf)^T}{2}$$ Then $$f(x)\approx f(x_0)+S\Delta x+A\Delta x$$ Could we regard $f(x_0)$ as a translation, $S\Delta x$ as an expansion or a contradiction along axes of eigen directions for it resembles strain tensor and $A\Delta x$ a rotation since skew-symmetric matrix is infinitesimal of rotation matrix?

If it is correct, I'm a bit confused with, say operator $A$, what object it acts on?

Otherwise, does this decomposition have any practical meaning in terms of vector field or velocity field?