Hi garrus. dx already answered the crux of your question but let's specialize to the case of ##\mathbb{R}^{n}##. Let ##U\subseteq \mathbb{R}^{n}## be open, ##p\in U##, and ##F:U\rightarrow \mathbb{R}^{m}## a map differentiable at ##p##. Recall that ##F## is differentiable at ##p## if there exists a linear map ##DF(p)## such that ##\lim _{v\rightarrow 0}\frac{|F(p + v) - F(p) - DF(p)v|}{|v|} = 0##. As you may remember, we call ##DF(p)## the total derivative of ##F## at ##p##. Now, as dx noted we may not be able to represent ##F## itself as a matrix if it isn't itself linear on ##\mathbb{R}^{n}## (in which case it agrees with its total derivative) but ##DF(p)## is linear and one can show that the standard matrix representation of ##DF(p)## is given by ##[DF(p)]_{S} = (\frac{\partial F^{j}}{\partial x^{i}}(p))##. This is none other than the Jacobian matrix. You can think of the linear map ##DF(p)## as being the best linear approximation of ##F## in a neighborhood of ##p##. As dx noted above, you can then develop such a formalism on arbitrary smooth manifolds.
