Noether Current: Understanding 2.10 & 2.11

[LCSphysicist]

Homework Statement

I can't see why the expression gives by the author is right.

Relevant Equations

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I just don't understand what happened after (2.11). That' is, the second term is zero, so we have
$$\alpha \Delta L = \alpha \partial_{\mu} ( \frac{\partial L}{\partial (\partial_{\mu}\phi)} \Delta \phi )$$
So, second (2.10), isn't $\Delta L = \alpha \partial_{\mu} J^{\mu}(x)$? So shouldn't the final equation reduce to this?:

$\alpha \alpha \partial_{\mu} J^{\mu}(x) = \alpha \partial_{\mu} ( \frac{\partial L}{\partial (\partial_{\mu}\phi)} \Delta \phi )$
$\partial_{\mu} (\frac{\partial L}{\partial (\partial_{\mu}\phi)} \Delta \phi - \alpha J^{\mu}(x) )$

 

[Gaussian97]

As defined in 2.11 $\Delta \mathcal{L} = \partial_\mu \mathcal{J}^\mu$, notice that the $\alpha$ factor is taken into account separatelly.