Confusion with multiple integrals represented by very different integral symbols

Question

This is a very basic (and possibly obvious) question but (to my knowledge) has not been clarified here yet. In vector/multivariable calculus, I see several notations being used. Using Gauss's law as an example, one notation is:

$$ \oint \mathbf{E} \cdot \mathbf{dA} = \frac{Q}{\epsilon_0} $$

Another notation is:

$$ \oint \mathbf{E} \cdot \mathbf{dS} = \frac{Q}{\epsilon_0} $$

Yet another notation is:

$$ \iint \mathbf{E} \cdot \mathbf{dS} = \frac{Q}{\epsilon_0} $$

And another notation is the same but with a loop on the integral (MathJax doesn't support it). There is even the notation:

$$ \int \mathbf{E} \cdot \mathbf{dA} = \frac{Q}{\epsilon_0} $$

What confuses to me is that all seem to refer to a surface integral of the electric field. But they respectively seem to mix and match notations for a line integral, a typical definite integral, a double integral, and a surface integral. Even more confusingly the RHS of Gauss's law can be written in all of the following notations:

$$ \frac{Q}{\epsilon_0} = \frac{1}{\epsilon_0} \int \rho dV = \frac{1}{\epsilon_0} \int \rho d^3 x = \frac{1}{\epsilon_0} \iiint \rho dV $$

So now there are 20 combinations with which to write Gauss's law! I assume they mean the same thing but they look like very different things. Do they actually mean the same thing? Is there a preferred notation? If so, why are there so many alternative notations?

Answer 1

TLDR:I would prefer the integral $∯_c$, which is called "surface integral" over a surface $c$.

For writing the equation given by Gauss's Law, you first want to write the flux at a certain surface element on the Gaussian surface due to a certain field, then sum it over all the fields, then sum it over all the elements.

Let's take this mathematically. Suppose we want to write the Gauss's Law equation for the most general scenario possible. We start with the simplest case of the field produced by a single charge $q$ inside the Gaussian surface.

$\sum_{dSϵC}$ $E_q.dS$ = $\frac{q}{ε_0}$

Okay. Now let us say we have multiple charges $q_1,q_2,q_3,....$ inside the Gaussian surface. We can write the equation for each charge $q_i$ as:

$\sum_{dSϵC}$ $E_{q_i}.dS$ = $\frac{q_i}{ε_0}$

When all those equations are summed together, you get:

$\sum_{q_i}$$\sum_{dSϵC}$ $E_{q_i}.dS$ = $\sum_{q_i}$$\frac{q_i}{ε_0}$

The order of the summations can be swapped to write

$\sum_{dSϵC}$$\sum_{q_i}$ $E_{q_i}.dS$ = $\sum_{q_i}$$\frac{q_i}{ε_0}$

In the case of charges $q_1',q_2'...$ present outside the Gaussian surface, they contribute zero electric flux and so we can modify the above equation to accomadate any charges present in the universe as:

$\sum_{dSϵC}$$\sum_{q_i}$ $E_{q_i}.dS$ + $\sum_{dSϵC}$$\sum_{q_i'}$ $E_{q_i'}.dS$ = $\sum_{dSϵC}$$\sum_{q_iϵuniverse}$ $E_{q_i}.dS$ = $\sum_{q_i}$$\frac{q_i}{ε_0}$

So, the Gaussian equation involving sigma notations should read:

$\sum_{dSϵC}$$\sum_{q_iϵuniverse}$ $E_{q_i}.dS$ = $\sum{\frac{q_{enclosed}}{ε_0}}$

Now supposing that the distribution of the charge is continuous and not discrete, it would be apt to replace $\sum_{q_iϵuniverse}$ $E_{q_i}$ with $\int E$. Furthermore the sum $\sum_{dSϵC}$ can be written in a much nicer way using a closed integral $∮_C$. Therefore, using integral notations,

$∮_C$$\int E.dS$ = $\sum{\frac{q_{enclosed}}{ε_0}}$

The double integration $∮_C$$\int$ may be denoted in a compact way by $∯_c$ and so

$∯_c$ $E.dS$ = $\sum{\frac{q_{enclosed}}{ε_0}}$, the ultimate Gauss Law equation.

In physics, I think the basic idea of the integral denoting the sum over all surface elements and all enclosed charge is pretty much universally understood while writing the Gauss's Law equation and so the different integral forms are sometimes used pretty loosely. But in terms of mathematical rigour, I think $∯_c$ is the best suited notation.

Hope this helps ^^