The integral form of Gauss' theorem

In summary, the Gauss theorem is often seen in the form of $$\frac{q}{\epsilon_0}=\oint\vec{E}d\vec{A}$$ which raises the question of why a line integral symbol is used for a surface integral. This is because in physics textbooks, the integral sign with a circle is used for both line integrals and surface integrals, with different notations for each. A proposed alternative notation is $$\frac{q}{\epsilon_0}=\oiint\vec{E}d\vec{A}$$ but it has not been widely adopted. This could explain why the symbol ##\oint## is used instead. Ultimately, it is important to follow the
  • #1
BearY
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In many texts I have seen, Gauss theorem has the form of$$\frac{q}{\epsilon_0}=\oint\vec{E}d\vec{A}$$
Why a line integral symbol was used for this surface integral everywhere? The more I see it the more I believe there is something wrong with my understanding about this.
I didn't think too much of this problem earlier, I remember I simply dismissed this question with it somehow being an application of the Stokes theorem. but now I am revisiting this question.
 
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  • #2
BearY said:
Why a line integral symbol was used for this surface integral everywhere?
An integral sign with a circle is commonly used (at least in physics textbooks) for both a line integral around a closed path, and a surface integral over a closed surface. The two kinds of integrals are usually distinguished by using something like ##d \vec l## for line integrals and ##d \vec a## for surface integrals.
 
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[tex]
\iint\hspace{-3.1ex}\bigcirc\ \vec E \cdot d\vec A
[/tex]
 

What is the integral form of Gauss' theorem?

The integral form of Gauss' theorem is a fundamental theorem in vector calculus that relates the flow of a vector field through a closed surface to the divergence of the vector field within the surface. It is also known as the divergence theorem or Gauss's flux theorem.

What is the significance of Gauss' theorem in physics?

Gauss' theorem is significant in physics because it allows for the calculation of the total flux of a vector field through a closed surface, which is useful in many physical applications such as electric and magnetic fields, fluid flow, and heat transfer.

What is the mathematical expression for Gauss' theorem?

The integral form of Gauss' theorem is expressed as ∫∫∫V(∇⋅F)dV = ∫∫S(F⋅n)dS, where V is the volume enclosed by the closed surface S, ∇⋅F is the divergence of the vector field F, and n is the unit normal vector to the surface at each point.

How is Gauss' theorem related to the divergence theorem?

Gauss' theorem is essentially the higher-dimensional version of the divergence theorem. Both theorems relate the flow of a vector field through a closed surface to the divergence of the vector field within the surface. However, the divergence theorem is limited to three dimensions, while Gauss' theorem can be applied in any number of dimensions.

What are some practical applications of Gauss' theorem?

Gauss' theorem has many practical applications in physics and engineering, such as calculating the electric flux through a closed surface due to a point charge, determining the rate of fluid flow through a pipe, and analyzing the flow of heat through a solid object. It is also used in mathematical modeling and computer simulations to solve complex problems involving vector fields.

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