- #1

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- 158

**Summary::**I am missing something in my integration by parts

Consider the infinitesimal variation of the fields ##\phi_a (x)##

$$\phi_a \rightarrow \phi_a + \delta \phi_a$$

The infinitesimal variation vanishes at the boundary of the region considered (ie. ##\delta \phi (x) = 0## at the boundary).

The definition of an action is

$$S = \int d^4 x \mathcal{L}(\phi, \partial_{\mu} \phi)$$

The infinitesimal variation of it is

$$\delta S = \int d^4 x \Big( \frac{\partial \mathcal{L}}{\partial \phi_a} \delta \phi_a + \frac{\partial \mathcal{L}}{\partial(\partial_{\mu} \phi_a)} \delta (\partial_{\mu} \phi_a)\Big)$$

Where I have used

$$\delta \phi_a = \frac{\partial \phi_a}{\partial x^{\mu}} \delta x^{\mu} = \partial_{\mu} \phi_a \delta x^{\mu}$$

Well I know that I have to integrate by parts here. What I have done is

But this is not the same that what Tong got

Note that I am missing ##\partial_{\mu}## in front of the second term. But I do not see why there should be one.

Thanks.

Reference: Tong's QFT Lecture notes, classical field theory, page 8.