# Riemann Manifold is not rectangular or spherical

1. Aug 13, 2013

### Philosophaie

I want to be able to formulate $$x^{n}$$ coordinate system.
$$x^{n} =(x^{1}, x^{2}, x^{3}, x^{4})$$
How do you do this when the Riemann Manifold is not rectangular or spherical?
Also how do you differentiate with respect to "s" in that case.
$$\frac{dx^n}{ds}$$

2. Aug 13, 2013

### ProfDawgstein

you can not simply do $\frac{dx^n}{ds}$, because you do not have a parameter $s$.

You should do something like this.
Make up a curve which is parameterized by $s$.
$s$ is your parameter along the curve.

Now you have

$x^{n}(s) = (x^{1}(s), x^{2}(s), x^{3}(s), x^{4}(s))$

And now you can do $\frac{dx^n}{ds}$ just fine.
Which is your tangent vector to the curve (might not be unit length).

For a 2d-surface you could use $u$ and $v$ as coordinates.