tensor math and coordinate systems 2 very basic questions on tensor math
1. when we refer to a curvy- linear coordinate system, do these systems (curvy linear coordinate systems) have an origin, where the value for each coordinate is zero, like in a cartesian coordinate sytem? or do the axes of the system never meet?
2. when we talk about transforming for one coordinate system to another, are we talking about transforming to another coordinate system on the same object or space, or are we jumping to a completely new object? 
thxs
 A: If it is not apparent by now, I'll give it away. These curvilinear coordinate systems are ones induced by embedded submanifolds $\Sigma \subset \mathbb{R}^N$ for some $N$. Therefore, given a point $p \in \Sigma$, we have a parametrization $(U, \phi) = (U, \phi^1,...,\phi^{\textbf{dim}(\Sigma):=k})$ which forms a coordinate system about a small neighborhood $V$ of $p$, why? Well given $q \in V$, there exists a unique $m_q$ such that $\phi(m_q) = q$ and so $q = (\phi^1(m_q),...,\phi^k(m_q))$. It is not the case that we need to have $\textbf{0} \in \Sigma$. However if you like, you can always modify this coordinate system to include $\textbf{0}$, by translating $\Sigma$. To do this we define another map $\psi(x) = \phi(q) - x$ where $x \in \phi(U)$. Thus the parametrizations $\{(U, \psi \circ \phi)\}$ give rise to a diffeomorphic copy of $\Sigma$ which contains the origin, and $\psi \circ \phi$ is the new coordinate system. 
The above answers all three questions. The fact that these systems come from embedded sub-manifolds, we now know that by curvilinear coordinates we mean coordinate systems in which coordinate-axes can be curved. Why? In the above, if we take any surface with positive curvature at $p$ (i.e $N = 3$), then one of the $\phi^j$-coordinate axes must be curved! I believe he other second question is whether you have to have $\textbf{0}$ to be a coordinate system, which is no!, by the above. However, we seen how one can get such a coordinate system from one which doesn't contain the origin. For the last question, when we are talking about changing coordinate systems we may be switching from,
$$(\phi^1,...,\phi^k) \to ((\psi \circ \phi)^1,..., (\psi \circ \phi)^k)$$
where the underlying space is still $\Sigma$ i.e the object as you describe it, doesn't change. However, consider $f: M^m \to N^n$ where $M^n = M, N^n = N$ are manifolds (the superscript denoting their dimension). Then $f$ induces a change in coordinates in the following sense, we know that there exists parametrizations $(U, \phi)$ and $(V, \psi)$ where $\phi: U \to M$ and $\psi: V \to N$. The locally we have that $f(\phi^1,...,\phi^m) = (\psi^1,...,\psi^n)$. We don't have to let $f$ be a homeomorphism or $M = N$ and so there underlying space (or object) does not have to be the same. 
