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

In summary: So in summary, the difference between the two relations is simply notational, with the sixth edition using ##\partial V## to represent the surface of the region, while the fifth edition uses ##S## for the same purpose. Additionally, the notation ##\partial V## is used to signify 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. In physics, ##\partial V## is often equivalent to ##S## in terms of representing the surface of the region.
  • #1
sams
Gold Member
84
2
In Mathematical Methods for Physicists, Sixth Edition, Page 60, Section 1.11, the Gauss' theorem is written as:
Gauss' Theorem.PNG

In Mathematical Methods for Physicists, Fifth Edition, Page 61, Section 1.11, the Gauss' theorem is written as:
Gauss' Theorem 2.jpeg

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...
 

Attachments

  • Gauss' Theorem.PNG
    Gauss' Theorem.PNG
    2.5 KB · Views: 855
  • Gauss' Theorem 2.jpeg
    Gauss' Theorem 2.jpeg
    3.7 KB · Views: 784
Last edited by a moderator:
Physics news on Phys.org
  • #2
sams said:
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##.
 
  • Like
Likes sams
  • #3
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.
 
  • Like
Likes dextercioby and sams
  • #5
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 $$v$$.
$$\partial v=S$$
 
  • Like
Likes sams
  • #6
Thank you all for your help and for your explanations
 

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

1. What is Gauss' Theorem?

Gauss' Theorem, also known as the Divergence Theorem, is a fundamental theorem in vector calculus that relates the surface integral of a vector field to the volume integral of its divergence.

2. Why are there two different notations used for Gauss' Theorem?

The two notations used for Gauss' Theorem are the integral form and the differential form. The integral form is used for calculations involving closed surfaces, while the differential form is used for calculations involving open surfaces. The choice of notation depends on the specific problem being solved.

3. How are the two notations related?

The two notations for Gauss' Theorem are mathematically equivalent and can be derived from each other. The integral form can be derived from the differential form by applying the Divergence Theorem, while the differential form can be derived from the integral form by using the fundamental theorem of calculus.

4. Which notation should I use?

The choice of notation for Gauss' Theorem depends on the problem at hand. If the surface is closed, the integral form should be used. If the surface is open, the differential form should be used. It is important to understand the differences between the two notations and choose the appropriate one for the specific problem.

5. What are some examples of when Gauss' Theorem is used?

Gauss' Theorem is used in many areas of physics and engineering, such as electromagnetism, fluid mechanics, and heat transfer. For example, it can be used to calculate the flux of an electric field through a closed surface or the flow rate of a fluid through an open surface. It is a powerful tool for solving problems involving vector fields and their behavior over surfaces.

Similar threads

B A question about Gauss' Theorem
    • General Math
Replies
2
Views
4K
A The δ Notation in Calculus of Variations
    • Calculus
Replies
2
Views
1K
Is this a good self-study program?
    • STEM Academic Advising
Replies
16
Views
537
MHB Understanding Theorem 3.4: Mean Value Theorem & Cauchy Riemann Equations
    • Topology and Analysis
Replies
2
Views
936
Div and curl in other coordinate systems
    • Calculus
Replies
2
Views
1K
Applied A question about mathematics textbooks for physicists
    • Science and Math Textbooks
Replies
28
Views
2K
Velocity is a vector in Newtonian mechanics
    • Mechanics
Replies
1
Views
947
Divergence Theorem in Curvilinear Coordinates: Questions & Explanations
    • Calculus and Beyond Homework Help
Replies
1
Views
897
Why are central force fields irrotational and conservative?
    • Mechanics
Replies
10
Views
2K
B Relating basis vectors at different points in a neighborhood
    • Differential Geometry
Replies
5
Views
1K

Hot Threads

Recent Insights

Back
Top