Need help understanding the concept of the Jacobian Matrix and its relation to differentiation As the question suggests, I need help understanding the concepts around the following differentiation of vector-valued functions:
1) The Norm. I understand that Norm to be defined as follows:
In the metric space defined on $\Bbb R^n$ to each vector $x = (x_1, x_2, ... , x_n)$ one can associate the nonnegative number $$\|x\| = d(x,0)$$ 
what on earth does this mean?
2) The Jacobian Matrix and its use in the definitions of differentiability and continuity. I understand the Jacobian matrix to be the gradient matrix i.e. a matrix whose elements are the partial derivatives of a function $f: \Bbb R^n \to \Bbb R^m$ at a point $(a,b) \in \Bbb R^n$. 
Now then a function is said to be differentiable if
 $$ \lim_{h \to 0} \frac{\|f(u+h) - f(u) - Ah\|}{\|h\|} = 0$$ where $u$ and $h$ are vectors in $\Bbb R^n$ and --this is where I'm perplexed -- $Ah = J_f(u)h \in \Bbb R^m$
What are we saying here? That $Ah$ is the derivative of said function? And are we saying that essentially a vector valued function is differentiated into a vector of its partial derivatives? 
 A: Don't worry, multidimensional analysis is a shock to the system the first time that you see it.

1) The norm is the distance to the origin $d(x,0)$, assumed to be Euclidean, so $$\lVert x \rVert = \sqrt{\sum x_i^2}$$

2) Okay, let's get a grip on this. $$f:\mathbb{R}^n \to \mathbb{R}^m$$ so we have functions $f_1(x_1,\cdots,x_n), \ldots, f_m(\cdots)$. There are really $m$ unrelated functions $f_i:\mathbb{R}^n \to \mathbb{R}$
Great; now, what is a derivative for a function like $f_i$? It's the gradient $\nabla f_i$. Hence the whole lot of derivative information is encoded in a $m\times n$ matrix $A=J_f(u)$ with components $$A_{ij} = \partial_j f_i$$
Right! But now we want a formal definition of differentiability. How do we do it for a single $f_i=g$? We say $\nabla g$ is the derivative if it tells us what the *directional * derivatives are. How does it do this?
We want $g(x+dx) = g(x) + dx \cdot \nabla g$ - small changes get dotted with the derivative. So formally, we want a vector $v$ such that $$\lim_{h\to 0} |g(x+h) - g(x) - h \cdot v | / \lVert h \rVert = 0$$ so that the error is smaller than $h$.
But now for all the $f_i$ we get a different vector $v_i$. Putting these all together into $A$ and choosing the denominator to bound the error for any one $f_i$ gives the result you have!
Edit: To summarize, $h$ is a small change in the coordinates, $Ah$ is the directional derivative of all the separate $f_i$s in this direction, and $A$ is the 'gradient' containing all derivative information.
A: Regarding the norm:
A norm is a different concept to a metric, although the two are related. 
A norm on a (real) vector space $V$ is a function $\|\cdot\|: V \rightarrow [0,\infty)$ satisfying:


*

*$\|\lambda u\| = |\lambda|\|u\|$, $\forall \lambda \in \mathbb{R}$

*$\|u\| = 0 \Leftrightarrow u = 0$ 

*$\|u + v\| \leq \|u\| + \|v\|$ (the triangle inequality)
Conceptually, a norm is a "length" - the norm of $u$ is like the "length" of $u$.
The definition in the question ($\|u\| := d(u,0)$) is not true for all metrics. For instance, the discrete metric (where the distance between two different points is always 1) will not satisfy the first property of a norm above. 
The converse is always possible though: for any normed vector space, we can define a metric $d(x,y) := \|x-y\|$ (which is called the metric induced by the norm)
For your second question, you're using a norm of elements of the space $\mathbb{R}^n$. Although there isn't one specific norm for a given space, but there are lots of results that you can use without actually knowing what the norm is, especially on a finite-dimensional space (in fact, any two norms on $\mathbb{R}^n$ can be shown to be equivalent).
