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
Spherical coordinates (r, θ, φ) as commonly used in physics: radial distance r, polar angle θ (theta), and azimuthal angle φ (phi). Source Wikipedia


Deducing the metric by the line element

In this Euclidian three-dimensionnal space, the line element is given by:

dl2 = dr2 + r22 + 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 = R22 + 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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