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How does one solve a problem like this?

Suppose we have

$$(e_\theta + f(\theta)e_\varphi) (e_\theta + f(\theta)e_\varphi)$$

What is the result of the above operation? As I remember it from the theory of covariant derivatives, the above relation would look like this

$$e_\theta[e_\theta] + e_\theta[f(\theta)e_\varphi] + f(\theta)e_\varphi[e_\theta] + f(\theta)e_\varphi[f(\theta)e_\varphi] = \nabla_\theta e_\theta + \nabla_\theta f(\theta)e_\varphi + f(\theta)\nabla_\varphi e_\theta + f(\theta)\nabla_\varphi f(\theta)e_\varphi$$

Now suppose the metric is Minkowskian, in which case all the ##\Gamma## vanishes. Then the last equality above would read

$$0 + \partial_\theta f(\theta) e_\varphi + 0 + 0 = \partial_\theta f(\theta) e_\varphi $$

Am I getting this correctly?

Suppose we have

$$(e_\theta + f(\theta)e_\varphi) (e_\theta + f(\theta)e_\varphi)$$

What is the result of the above operation? As I remember it from the theory of covariant derivatives, the above relation would look like this

$$e_\theta[e_\theta] + e_\theta[f(\theta)e_\varphi] + f(\theta)e_\varphi[e_\theta] + f(\theta)e_\varphi[f(\theta)e_\varphi] = \nabla_\theta e_\theta + \nabla_\theta f(\theta)e_\varphi + f(\theta)\nabla_\varphi e_\theta + f(\theta)\nabla_\varphi f(\theta)e_\varphi$$

Now suppose the metric is Minkowskian, in which case all the ##\Gamma## vanishes. Then the last equality above would read

$$0 + \partial_\theta f(\theta) e_\varphi + 0 + 0 = \partial_\theta f(\theta) e_\varphi $$

Am I getting this correctly?

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