4d integration/differentiation notation and the total derivative

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

1. What is 4d integration/differentiation notation?

4d integration/differentiation notation is a mathematical notation used to represent the process of finding the integral or derivative of a function in four-dimensional space. It is similar to the traditional notation used in three-dimensional space, but it includes an additional variable, typically represented by the letter "w".

2. How is 4d integration/differentiation notation used in scientific research?

4d integration/differentiation notation is commonly used in scientific research to analyze and model complex systems in four-dimensional space, such as fluid dynamics, electromagnetism, and quantum mechanics. It allows scientists to better understand the behavior of these systems and make predictions about their future behavior.

3. What is the difference between 4d integration/differentiation notation and traditional notation?

The main difference between 4d integration/differentiation notation and traditional notation is the inclusion of an additional variable "w" in the former. This extra variable represents the fourth dimension and allows for the analysis of systems in four-dimensional space. Traditional notation only includes three variables, typically represented by "x", "y", and "z".

4. What is the total derivative in 4d integration/differentiation notation?

The total derivative in 4d integration/differentiation notation is a mathematical concept that represents the change in a function with respect to all of its variables, including the fourth dimension "w". It is a generalization of the traditional partial derivative, which only considers changes in the function with respect to one variable at a time.

5. How is the total derivative used in scientific calculations?

The total derivative is used in scientific calculations to determine the rate of change of a function in four-dimensional space. It is particularly useful in physics and engineering, where systems often involve multiple variables that are interdependent. By calculating the total derivative, scientists can gain a better understanding of how these variables affect the overall behavior of the system.

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