# 4d integration/differentiation notation and the total derivative

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• binbagsss
In summary, The notation of a 4D integral is ##d^4x=dx^{\nu}## and when considering a total derivative, the 4th order integral becomes a regular 3D integral of a function. The index "mu" is a dummy free index, meaning it can only have one value and the 4th order integral becomes a 3D integral of a function. Therefore, there is no need for a ##\delta_{\nu}^{\mu}## term in this case.

#### binbagsss

TL;DR Summary
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?

"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$$\int_{x_{1,i}}^{x_{1,f}} dx_1 {} \int_{x_{2,i}}^{x_{2,f}} dx_2 {} \int_{x_{3,i}}^{x_{3,f}} dx_3 {} [\phi(x_{0,f},x_1,x_2,x_3)-\phi(x_{0,i},x_1,x_2,x_3)]$$