In this article, we will calculate the Euclidian metric tensor for a surface of a sphere in spherical coordinates by two ways, as seen in the previous article Generalisation of the metric tensor
- - By deducing the metric directly from the space line element
- - By calculating the metric from the product of derivatives of the two-dimensional Cartesian coordinates system
Deducing the metric by the line element
In this Euclidian three-dimensionnal space, the line element is given by:
dl2 = dr2 + r2dθ2 + r2sin2θdΦ2
If we set the polar coordinate r to be some constant R we lose the dr term (because r is now constant) and the line element now becomes:
dl2 = R2dθ2 + R2sin2θdΦ2
which describes a two-dimensional surface using the two polar coordinates (θ, Φ)
Or we know from the previous article that this line element could be written as:
dl2 = gijdxidyj
We can deduce immediately that the metric and inverse metric for this surface, using coordinates x0=θ and x1=Φ, are:
This was the easy part. Let's try to calculate the same metric by using the formula of the coordinates derivatives product.
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