This is probably a basic question. I'm actually taking a class that introduces me to Maxwell's equations. I am currently trying to make sense of the Gauss's law and have some difficulty understanding the flux of a vector field.

The book first starts by explaining the surface integral of a scalar field, using this:

$$ M = \int_S \sigma(x, y)~\mathrm{d} a $$

where $\delta a$ is a infinitesimal area of the surface and $\sigma$ a function returning the area mass density. So far so good for me.

But then it goes off explaining the flux of a vector field:

$$ M = \int_S \vec{A} \cdot \hat{n}~\mathrm{d} a $$

But now this section starts saying that this integral is over a vector field. Which I don't really understand since the dot product of two vectors gives back a scalar. What do I get wrong here?

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    $\begingroup$ You really get nothing wrong here. This is how the flux is defined. In the end you still integrate over a vectorfield since $n$ comes from the surface. Just the result you are getting is a scalar. Does this answer your question? $\endgroup$ – tomtom1-4 Oct 17 at 17:21
  • $\begingroup$ I would suggest considering the problem for simple surfaces. Pick a simple vector field and a simple surface, e.g. a plane, the surface of a cube, and do the integral with pen and paper. You will need to find dot product on each surface and then add. This will help you to get a feeling for what is happening. Good luck $\endgroup$ – Cryo Oct 17 at 17:38
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    $\begingroup$ As @tomtom1-4 said, you are really integrating over a scalar field, not a vector field. What does the book literally say? Maybe you misread something. $\endgroup$ – S V Oct 17 at 18:26
  • $\begingroup$ Keep in mind that the unit vector “n” is defined as being perpendicular to the surface. With the dot product, you are finding the component of the field vector which is parallel to “n” and perpendicular to the surface. For flux, you are looking for the amount of field “crossing” the surface. $\endgroup$ – R.W. Bird Oct 17 at 18:27
  • $\begingroup$ Thank you, guys. It appears I misread the introduction for the flux. It says "In Gauss’s law, the surface integral is applied not to a scalar function (such as the density of a surface) but to a vector field", but further down the road, it says the result is a scalar. $\endgroup$ – André Jacques Oct 17 at 19:22

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