Formulating x^n Coordinate System for Non-Rectangular/Spherical Riemann Manifold

[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}

 

[ProfDawgstein]

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}

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).

How do you do this when the Riemann Manifold is not rectangular or spherical?

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

What do you want to do?