Metric tensor exercise: calculation for the surface of a sphere

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 = g_ijdxⁱdy^j

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.