I 4d integration/differentiation notation and the total derivative

[binbagsss]

TL;DR

4d notation integration/differentiation

This is probably a stupid question but,

$\frac{d\partial_p}{d\partial_c}=\delta^p_c$

For the notation of a 4D integral it is $d^4x=dx^{\nu}$, so if I consider a total derivative:

$\int\limits^{x_f}_{x_i} \partial_{\mu} (\phi) d^4 x = \phi \mid^{x_f}_{x_i}$

why is there no $\delta_{\nu}^{\mu}$ sort of term required?

 

[dextercioby]

"Mu" is a dummy free index, so it is imbalanced. You can only pick one value for it and integrate with respect to it. Therefore, the 4th order integral becomes a regular 3d integral of a function. Let us differentiate with respect to $x_0$. We obtain$$\int_{x_{1,i}}^{x_{1,f}} dx_1 {} \int_{x_{2,i}}^{x_{2,f}} dx_2 {} \int_{x_{3,i}}^{x_{3,f}} dx_3 {} [\phi(x_{0,f},x_1,x_2,x_3)-\phi(x_{0,i},x_1,x_2,x_3)] $$