TITLE: Taylor Series for functions $f:R^n\rightarrow R^n$
QUESTION [8 upvotes]: I've been told that we can write a taylor series for functions $f:R^n\rightarrow R$ but we can't write one for $f:R^n\rightarrow R^n$.  I'm not quite sure why this not possible, but I suspect it have something to do with the mean value theorem.  Could anyone shed some light on this?

REPLY [3 votes]: The same formula that applies for scalar fields applies for vector fields.
$$f(x) = f(a) + [(x-a) \cdot \nabla] f \Big|_a + \frac{1}{2!} [(x-a) \cdot \nabla]^2 f \Big |_a + \ldots$$
$f$ can be scalar or vector.