What are the partial derivatives of a map $f:\mathbb{R}^n \rightarrow \mathbb{R}^m$ I was wondering what the definition of a partial derivative of such a map is? 
Is it $\partial_if_j$ or is it anything else? The reason why I have doubts is that I found the definition that a partial derivative is the total derivative of a function $f(\,\cdot\,,x_2): x_1 \mapsto f(x_1,x_2)$, where $(x_1,x_2)$ is some decomposition of $x \in \mathbb{R}^n$. 
 A: The most natural definition would be this, I think: A map $f: \mathbb{R}^n \to \mathbb{R}^m$ can always be decomposed into $m$ maps $f_1, f_2, \ldots, f_m : \mathbb{R}^n \to \mathbb{R}$ that satisfy
$$f(x) = (f_1(x), f_2(x), \ldots, f_m(x))$$
Then one can define the partial derivative of $f$ with respect to the $i$th argument simply by
$$\frac{\partial f}{\partial x_i}(x) = \left( \frac{\partial f_1}{\partial x_i}(x), \frac{\partial f_2}{\partial x_i}(x), \ldots, \frac{\partial f_m}{\partial x_i}(x) \right)$$
The Jacobian matrix of $f$ is formed by letting these "partial derivatives" be column vectors in an $m \times n$ matrix.
A: You already accepted an answer, but there is an even simpler and more general way to look at it, if you know what a topological vector space is (for example, any vector space with a norm is a topological vector space).  If $S$ is a (real) topological vector space, and $f:\mathbb{R}^n \to S$, then define
$$ \frac{\partial f}{\partial x_i}(\mathbf{x}) = 
\lim_{h \to 0}\frac{f(\mathbf{x}+h\mathbf{e}_i)- f(\mathbf{x})}{h},  $$
if the limit exists.  Here $\mathbf{e}_i$ is the unit vector in $\mathbf{R}^n$ with $1$ in the $i$th coordinate and $0$ in all the other coordinates.  If $S = \mathbf{R}^m$, this is equivalent to Svinepels's answer.
I do not know how much, if at all, this can be generalized to a more general algebraic structure than "topological vector space".
A: It helps a lot to think the derivative in matrix terms as   $$Jf=\left[\begin{array}{cccc}    
{\partial f_1\over\partial x_1}&{\partial f_1\over\partial x_2}&\cdots&{\partial f_1\over\partial x_n}\\
{\partial f_2\over\partial x_1}&{\partial f_2\over\partial x_2}&\cdots&{\partial f_2\over\partial x_n}\\
\vdots&&\ddots&\vdots\\
{\partial f_m\over\partial x_1}&{\partial f_m\over\partial x_2}&\cdots&{\partial f_m\over\partial x_n}
\end{array}\right],$$ where you can see in each row the gradients of each component $f_i$. The reason to use the "name" $Jf$, it is because it is also dubbed as the "Jacobian matrix" of $f$. With this device, it becomes more transparent how is the chain's rule and many other things.
