Gauss' Theorem -- Why two different notations are used?

[sams]

In Mathematical Methods for Physicists, Sixth Edition, Page 60, Section 1.11, the Gauss' theorem is written as:

In Mathematical Methods for Physicists, Fifth Edition, Page 61, Section 1.11, the Gauss' theorem is written as:

Kindly I would like to know please:
1. What is the difference between the two relations?
2. What does $\partial{V}$ in Equation (1.101a) stands for? In physics, I realized that $\partial{V}$ is usually not included when Gauss' theorem is used, why is that?

Thanks a lot for your help...

 

[PeroK]

In Mathematical Methods for Physicists, Sixth Edition, Page 60, Section 1.11, the Gauss' theorem is written as:
View attachment 231423
In Mathematical Methods for Physicists, Fifth Edition, Page 61, Section 1.11, the Gauss' theorem is written as:
View attachment 231424
Kindly I would like to know please:
1. What is the difference between the two relations?
2. What does $\partial{V}$ in Equation (1.101a) stands for? In physics, I realized that $\partial{V}$ is usually not included when Gauss' theorem is used, why is that?

Thanks a lot for your help...

Do these books not make their notation clear? The only difference is notational.

$\partial V$ is sometimes used for the surface of a region $V$. In the second equation, simply $S$ is used for the surface of the region $V$.

 

[jedishrfu]

The notation used by the sixth edition was to remind the reader that the left hand side is a double integral over the surface of the region and the right hand side is a triple integral over the volume of the region. They likely changed it as someone brought it to their attention or an editor schooled as a physicist took issue and decided it was best to change it.

 

[Svein]

https://en.wikipedia.org/wiki/Stokes'_theorem at the end of the page.

 

[cristianbahena]

Usually un physics triple or double integrals $$\int \int$$ $$\int \int \int$$
Are changed by only one simbol $$\int$$, so that is the same equation.

The simbol $$\partial v$$ means that the integral Is computed on boundary superfice of $$v$$ or on boundary of $$v$$.
$$\partial v=S$$

 

[sams]

Thank you all for your help and for your explanations