Differential geometry: coordinate patches

In summary, the conversation discusses how to show that u^1 is arc length on the u^1 curves if g_{11} \equiv 1. The attempt at a solution involves using the formula for arc length \frac{ds}{dt} = \sum g_{ij} \frac {d\alpha^{i}}{dt} \frac {d\alpha^{j}}{dt} and the metric g_{ij}(u^{1}, u^{2})= <x_{i}(u^{1}, u^{2}), x_{j}(u^{1}, u^{2}) to show that u^1 must be arc length. However, the solution is not fully completed and further help is needed.
  • #1
SNOOTCHIEBOOCHEE
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Homework Statement



For a coordinate patch x: U--->[tex]\Re^{3}[/tex]show that[tex]u^{1}[/tex]is arc length on the [tex]u^{1}[/tex] curves iff [tex]g_{11} \equiv 1[/tex]

The Attempt at a Solution



So i know arc legth of a curve [tex]\alpha (t) = \frac{ds}{dt} = \sum g_{ij} \frac {d\alpha^{i}}{dt} \frac {d\alpha^{j}}{dt}[/tex] (well that's actually arclength squared but whatever).

But I am not sure how to write this for just a [tex]u^{1}[/tex] curve. A [tex]u^{1}[/tex] curve throught the point P= x(a,b) is [tex]\alpha(u^{1})= x(u^{1},b)[/tex]

But i have no idea how to find this arclength applies to u^1 curves.

Furthermore i know some stuff about our metric [tex]g_{ij}(u^{1}, u^{2})= <x_{i}(u^{1}, u^{2}), x_{j}(u^{1}, u^{2})[/tex]

But i do not know how to use that to show that u^1 must be arclength but here is what i have so far:

[tex]g_{11}(u^{1}, b)= <x_{1}(u^{1}, u^{2}), x_{2}(u^{1}, u^{2})>[/tex] We know that [tex]x_{1}= (1,0)[/tex] and that is as far as i got :/

Any help appreciated.
 
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  • #2
bump, i still need help on this
 
  • #3
one last bump, can anybody help me on this?
 

What is a coordinate patch in differential geometry?

A coordinate patch in differential geometry is a subset of a manifold where a coordinate system can be defined. It is used to break down a complex surface into smaller, more manageable parts in order to better understand its geometric properties.

How are coordinate patches used in differential geometry?

Coordinate patches are used to define a local coordinate system on a manifold, allowing for the measurement of distances and angles between points on the surface. They also allow for the calculation of geometric properties, such as curvature and torsion, which are essential in understanding the behavior of a surface.

What is the purpose of using multiple coordinate patches?

Multiple coordinate patches are used to cover a manifold in order to fully describe its geometric properties. This is because it is not always possible to define a single coordinate patch that covers the entire surface without any distortions or overlaps.

What is the difference between a coordinate patch and a chart in differential geometry?

A coordinate patch refers to the actual subset of a manifold where a coordinate system is defined, while a chart refers to the coordinate system itself. A chart is a one-to-one mapping from a coordinate patch to a set of real numbers, allowing for the measurement of distances and angles on the surface.

How do coordinate patches relate to the concept of differentiability in differential geometry?

The use of coordinate patches is essential in defining the concept of differentiability on a manifold. By breaking down a surface into smaller parts, differentiability can be defined at each patch, which then allows for the calculation of derivatives and other geometric properties.

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