In page 14 to get the equation of motion, it takes the variation of the action $$ S_P[X,\gamma]=-\frac{1}{4\pi\alpha'}\int_Md\tau d\sigma(-\gamma)^{1/2}\gamma^{ab}\partial_a X^\mu\partial_b X_\mu $$ and get the equation (1.2.27) $$ \delta S_P=\frac{1}{2\pi\alpha'}\int_{-\infty}^{\infty}d\tau\int_0^ld\sigma(-\gamma)^{1/2}\delta X^\mu\nabla^2X_\mu\\ -\frac{1}{2\pi\alpha'}\int_{-\infty}^{\infty}d\tau(-\gamma)^{1/2}\delta X^\mu\partial^\sigma X_\mu|_{\sigma=0}^{\sigma=l}. $$ He took the coordinate region to be $$ -\infty<\tau<\infty, 0\leq\sigma\leq l $$ and said this is a single string propagating without sources.

I cannot understand why he dropped the $\tau$ term such as terms $\partial^\tau X_\mu$ and only leave the $\sigma$ term. Could someone help me?


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