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In the second edition of QFT by Mandl & Shaw, for the unnumbered eqn below eqn (13.120c) on page 297, I get a factor of -1 on the rhs that the book doesn't have. However, in the next two equations, it is clear that the author intends for the eqn to stand as it is. I get:

[tex]F_{\mu\nu}F^{\mu\nu} = (\partial_{\nu}A_{\mu} - \partial_{\mu}A_{\nu})(\partial^{\nu}A^{\mu} - \partial^{\mu}A^{\nu})[/tex]

[tex]= \partial_{\nu}A_{\mu}\partial^{\nu}A^{\mu} - \partial_{\mu}A_{\nu}\partial^{\nu}A^{\mu} - \partial_{\nu}A_{\mu}\partial^{\mu}A^{\nu} + \partial_{\mu}A_{\nu}\partial^{\mu}A^{\nu}[/tex]

[tex] = (\partial_{\nu}A_{\mu})(\partial^{\nu}A^{\mu}) + A_{\mu}\Box A^{\mu} - (\partial_{\mu}A_{\nu})(\partial^{\nu}A^{\mu}) - A_{\nu}\partial_{\mu}\partial^{\nu}A^{\mu}

- (\partial_{\nu}A_{\mu})(\partial^{\mu}A^{\nu}) - A_{\mu}\partial_{\nu}\partial^{\mu}A^{\nu} + (\partial_{\mu}A_{\nu})(\partial^{\mu}A^{\nu}) + A_{\nu}\Box A^{\nu}[/tex]

[tex] = 2A_{\mu}\Box A^{\mu} - 2A^{\mu}\partial_{\mu}\partial_{\nu}A^{\nu} + 2(\partial_{\nu}A_{\mu})F^{\mu\nu}[/tex]

[tex]\int d^4 x(\partial_{\nu}A_{\mu})F^{\mu\nu} = -\int d^4 x A_{\mu}\partial_{\nu}F^{\mu\nu} = 0[/tex]

using eqn (5.2) on page 74 and s = 0. So I get an extra minus sign on the rhs.

[tex]-\frac{1}{4}\int d^4x F_{\mu\nu}F^{\mu\nu} = -\frac{1}{2} \int d^4 x A^{\mu}[g_{\mu\nu}\Box - \partial_{\mu}\partial_{\nu}]A^{\nu}[/tex]

What am I doing wrong?

[tex]F_{\mu\nu}F^{\mu\nu} = (\partial_{\nu}A_{\mu} - \partial_{\mu}A_{\nu})(\partial^{\nu}A^{\mu} - \partial^{\mu}A^{\nu})[/tex]

[tex]= \partial_{\nu}A_{\mu}\partial^{\nu}A^{\mu} - \partial_{\mu}A_{\nu}\partial^{\nu}A^{\mu} - \partial_{\nu}A_{\mu}\partial^{\mu}A^{\nu} + \partial_{\mu}A_{\nu}\partial^{\mu}A^{\nu}[/tex]

[tex] = (\partial_{\nu}A_{\mu})(\partial^{\nu}A^{\mu}) + A_{\mu}\Box A^{\mu} - (\partial_{\mu}A_{\nu})(\partial^{\nu}A^{\mu}) - A_{\nu}\partial_{\mu}\partial^{\nu}A^{\mu}

- (\partial_{\nu}A_{\mu})(\partial^{\mu}A^{\nu}) - A_{\mu}\partial_{\nu}\partial^{\mu}A^{\nu} + (\partial_{\mu}A_{\nu})(\partial^{\mu}A^{\nu}) + A_{\nu}\Box A^{\nu}[/tex]

[tex] = 2A_{\mu}\Box A^{\mu} - 2A^{\mu}\partial_{\mu}\partial_{\nu}A^{\nu} + 2(\partial_{\nu}A_{\mu})F^{\mu\nu}[/tex]

[tex]\int d^4 x(\partial_{\nu}A_{\mu})F^{\mu\nu} = -\int d^4 x A_{\mu}\partial_{\nu}F^{\mu\nu} = 0[/tex]

using eqn (5.2) on page 74 and s = 0. So I get an extra minus sign on the rhs.

[tex]-\frac{1}{4}\int d^4x F_{\mu\nu}F^{\mu\nu} = -\frac{1}{2} \int d^4 x A^{\mu}[g_{\mu\nu}\Box - \partial_{\mu}\partial_{\nu}]A^{\nu}[/tex]

What am I doing wrong?

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