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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:

**dl ^{2} = dr^{2} + r^{2}dθ^{2} + r^{2}sin^{2}θdΦ^{2}**

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:

**dl ^{2} = R^{2}dθ^{2} + R^{2}sin^{2}θdΦ^{2}**

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:

**dl ^{2} = g_{ij}dx^{i}dy^{j}**

We can deduce immediately that the metric and inverse metric for this surface, using coordinates x^{0}=θ and x^{1}=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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