In the precedent article Covariant differentiation exercise 1: calculation in cylindrical coordinates, we have deduced the expression of the covariant derivative of a tensor of rank 1, i.e of a contravariant vector - type (1,0) or of a covariant vector - type (0,1).

It can be shown that the covariant derivatives of higher rank tensors are constructed from the following building blocks:

- - take the partial derivatives of the tensor
- - add a Γ
^{α}_{γβ}term for each upper index - - substract a Γ
^{γ}_{αβ}term for each lower index

Following the three rules given above, we obtain for tensors of rank 2, respectively of type (1,1) T^{μ}_{ν} , (0,2) T_{μν} and (2,0) T^{μν}:

We recall from our article Minkowski's Four-Dimensional Space-Time the Euclidean metric tensor's expression for Cartesian coordinates

And substituting g_{ij} in the second above equality dedicated to type (0,2) tensor gives

As all the terms g_{ij} are constant, the first term δg_{ij}/δx^{β} is null. And because there is no curvature in the Euclidean space, all the Christoffel symbols vanish, so that the conplete right-hand side of the equation equals zero, and therefore

But this is the magic and clever thing about tensors, **if this equation holds true in a particular coordinates system, here the Cartesian coordinates, it must be true for the Euclidean metric in ALL coordinate systems**.

##### Covariant derivation of the Euclidean metric in spherical coordinates

Let's try to verify this by calculating one component of the covariant differentiation in the spherical coordinates.

We recall from our article that in spherical coordinates, the metric's expression is

If we were to calculate the component g_{ΦΦ;θ}, we should then write

But g_{αφ} !=0 only if α = φ, based on the above expression, so we can simplify this equation

Or

And from previous article Christoffel symbol exercise: calculation in polar coordinates part II, we know the expression of the Christoffel symbols Γ^{Φ}_{θΦ} = Γ^{Φ}_{Φθ} = cosθ/sinθ

so that

Finally, we confirm that this component of the covariant derivative with respect to θ equals also zero in a polar coordinates system, as expected

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