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Let us suppose we have a covariant derivative of a contravariant vector such as

$$\nabla_{\mu}V^{\nu}=\partial_{\mu}V^{\nu} + \Gamma^{\nu}_{\mu \lambda}V^{\lambda}$$

If ##\Delta_{\mu}V^{\nu}## is a (1,1) Tensor, it must be transformed as

$$\nabla_{\bar{\mu}}V^{\bar{\nu}} = \frac{ \partial x^{\bar{\nu}}}{\partial x^\nu}\frac{\partial x^{\mu}}{\partial x^{\bar{\mu}}}\nabla_{\mu}V^{\nu}$$

Or

$$\nabla_{\bar{\mu}}V^{\bar{\nu}} = \frac{ \partial x^{\bar{\nu}}}{\partial x^\nu}\frac{\partial x^{\mu}}{\partial x^{\bar{\mu}}}[\partial_{\mu}V^{\nu} + \Gamma^{\nu}_{\mu \lambda}V^{\lambda}]$$

Now my question is does ##\frac{ \partial x^{\bar{\nu}}}{\partial x^\nu}\frac{\partial x^{\mu}}{\partial x^{\bar{\mu}}}## acts a differential or it acts like a multiplication on the ##[\partial_{\mu}V^{\nu} + \Gamma^{\nu}_{\mu \lambda}V^{\lambda}]## ?

So which of these expressions is true

1) $$\frac{ \partial x^{\bar{\nu}}}{\partial x^\nu}\frac{\partial x^{\mu}}{\partial x^{\bar{\mu}}}(\partial_{\mu})V^{\nu} + \frac{ \partial x^{\bar{\nu}}}{\partial x^\nu}\frac{\partial x^{\mu}}{\partial x^{\bar{\mu}}}(V^{\nu} )\partial_{\mu} + \frac{ \partial x^{\bar{\nu}}}{\partial x^\nu}\frac{\partial x^{\mu}}{\partial x^{\bar{\mu}}}(\Gamma^{\nu}_{\mu \lambda}) V^{\lambda}+ \frac{ \partial x^{\bar{\nu}}}{\partial x^\nu}\frac{\partial x^{\mu}}{\partial x^{\bar{\mu}}}(V^{\lambda})\Gamma^{\nu}_{\mu \lambda}$$

2)$$\frac{ \partial x^{\bar{\nu}}}{\partial x^\nu}\frac{\partial x^{\mu}}{\partial x^{\bar{\mu}}}\partial_{\mu}V^{\nu} + \frac{ \partial x^{\bar{\nu}}}{\partial x^\nu}\frac{\partial x^{\mu}}{\partial x^{\bar{\mu}}}\Gamma^{\nu}_{\mu \lambda}V^{\lambda}$$

In principle the correct equation need to be 1 I guess but I cannot be sure somehow.

I am was doing inverse operations of these equations (showing that ##\nabla_{\bar{\mu}}V^{\bar{\nu}}## transforms as a tensor) and I have obtained something like this

There are extra two terms which I cannot explain why are they. The only logical explanation seemed to be that the ##\frac{ \partial x^{\bar{\nu}}}{\partial x^\nu}\frac{\partial x^{\mu}}{\partial x^{\bar{\mu}}}## must act as a differential operation on ##[\partial_{\mu}V^{\nu} + \Gamma^{\nu}_{\mu \lambda}V^{\lambda}]##

$$\nabla_{\mu}V^{\nu}=\partial_{\mu}V^{\nu} + \Gamma^{\nu}_{\mu \lambda}V^{\lambda}$$

If ##\Delta_{\mu}V^{\nu}## is a (1,1) Tensor, it must be transformed as

$$\nabla_{\bar{\mu}}V^{\bar{\nu}} = \frac{ \partial x^{\bar{\nu}}}{\partial x^\nu}\frac{\partial x^{\mu}}{\partial x^{\bar{\mu}}}\nabla_{\mu}V^{\nu}$$

Or

$$\nabla_{\bar{\mu}}V^{\bar{\nu}} = \frac{ \partial x^{\bar{\nu}}}{\partial x^\nu}\frac{\partial x^{\mu}}{\partial x^{\bar{\mu}}}[\partial_{\mu}V^{\nu} + \Gamma^{\nu}_{\mu \lambda}V^{\lambda}]$$

Now my question is does ##\frac{ \partial x^{\bar{\nu}}}{\partial x^\nu}\frac{\partial x^{\mu}}{\partial x^{\bar{\mu}}}## acts a differential or it acts like a multiplication on the ##[\partial_{\mu}V^{\nu} + \Gamma^{\nu}_{\mu \lambda}V^{\lambda}]## ?

So which of these expressions is true

1) $$\frac{ \partial x^{\bar{\nu}}}{\partial x^\nu}\frac{\partial x^{\mu}}{\partial x^{\bar{\mu}}}(\partial_{\mu})V^{\nu} + \frac{ \partial x^{\bar{\nu}}}{\partial x^\nu}\frac{\partial x^{\mu}}{\partial x^{\bar{\mu}}}(V^{\nu} )\partial_{\mu} + \frac{ \partial x^{\bar{\nu}}}{\partial x^\nu}\frac{\partial x^{\mu}}{\partial x^{\bar{\mu}}}(\Gamma^{\nu}_{\mu \lambda}) V^{\lambda}+ \frac{ \partial x^{\bar{\nu}}}{\partial x^\nu}\frac{\partial x^{\mu}}{\partial x^{\bar{\mu}}}(V^{\lambda})\Gamma^{\nu}_{\mu \lambda}$$

2)$$\frac{ \partial x^{\bar{\nu}}}{\partial x^\nu}\frac{\partial x^{\mu}}{\partial x^{\bar{\mu}}}\partial_{\mu}V^{\nu} + \frac{ \partial x^{\bar{\nu}}}{\partial x^\nu}\frac{\partial x^{\mu}}{\partial x^{\bar{\mu}}}\Gamma^{\nu}_{\mu \lambda}V^{\lambda}$$

In principle the correct equation need to be 1 I guess but I cannot be sure somehow.

I am was doing inverse operations of these equations (showing that ##\nabla_{\bar{\mu}}V^{\bar{\nu}}## transforms as a tensor) and I have obtained something like this

There are extra two terms which I cannot explain why are they. The only logical explanation seemed to be that the ##\frac{ \partial x^{\bar{\nu}}}{\partial x^\nu}\frac{\partial x^{\mu}}{\partial x^{\bar{\mu}}}## must act as a differential operation on ##[\partial_{\mu}V^{\nu} + \Gamma^{\nu}_{\mu \lambda}V^{\lambda}]##

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