Differential Geometry¶
Metric¶
Definitions¶
Denote the basis in use as ˆeμ, then the metric can be written as
if the basis satisfies
Inversed metric
How to calculate the metric¶
Let’s check the definition of metric again.
If we choose a basis ˆeμ, then a vector (at one certain point) in this coordinate system is
Then we can construct the expression of metric of this point under this coordinate system,
For example, in spherical coordinate system,
Now we have to find the basis under spherical coordinate system. Assume the basis is ˆer,ˆeθ,ˆeϕ. Choose some scale factors hr=1,hθ=r,hϕ=rsinθ. Then the basis is
etc. Then collect the terms in formula (6) is we get →x=rˆer, this is incomplete. So we check the derivative.
Once we reach here, the component (er,eθ,eϕ) of the point under the spherical coordinates system basis (ˆer,ˆeθ,ˆeϕ) at this point are clear, i.e.,
In this way, the metric tensor for spherical coordinates is
Connection¶
First class connection can be calculated
Second class connection isfootnote{Kevin E. Cahill}
Gradient, Curl, Divergence, etc¶
Gradient¶
Divergence¶
For an anti-symmetric tensor
Annotation Using the relation g=gμνAμν, Aμν is the algebraic complement, we can prove the following two equalities.
In some simple case, all the three kind of operation can be demonstrated by different applications of the del operator, which ∇≡ˆx∂x+ˆy∂y+ˆz∂z.
- Gradient, ∇f, in which f is a scalar.
- Divergence, ∇⋅→v
- Curl, ∇×→v
- Laplacian, Δ≡∇⋅∇≡∇2