Integral form of Maxwell equations.

[space-time]

I have been studying the Maxwell equations recently (namely the integral forms of them). Of course I had to study line integrals before that. Well, I went to a hyperphysics page to look up the equations:

http://hyperphysics.phy-astr.gsu.edu/hbase/electric/maxeq.html

I noticed that the notations for those integrals do not match the notations for line integrals of vector fields that I learned on the following page:

http://tutorial.math.lamar.edu/Classes/CalcIII/CurlDivergence.aspx

(Look in the last two blue boxes on the page I just posted)

The integrals in the Maxwell equations on the hyperphysics page have differentials (dA and ds) that have the vector arrow head symbols over them.

In the line integrals on the Calculus III page (http://tutorial.math.lamar.edu/Classes/CalcIII/CurlDivergence.aspx) in the last two blue boxes, you can see that the differentials dA and ds are present either in the line integral itself or in the double integral. However, on this page those differentials do not have arrow heads over them.This makes me question whether the integrals in the Maxwell equations are actually even line integrals or if they are surface integrals. I wonder this because I notice that on this page on surface integrals (http://tutorial.math.lamar.edu/Classes/CalcIII/SurfIntVectorField.aspx) the integrals have the differential ds with an arrow head over it (unlike on the page with the line integrals). That aspect is similar to some of the integrals in the Maxwell equations on the hyperphysics page. However, the ds (w/ arrow head) differential is only present within double integrals on the page about surface integrals (unlike in the hyperphysics page where the ds w/ arrow head differential is present in integrals that look like line integrals that satisfy Green's theorem.) Additionally, I haven't seen any surface integrals that have the circle in the middle of the integral sign like the ones on the hyperphysics page. I have only seen that on line integrals. Finally, no dA (w/ arrowhead) differential appears on the surface integral page. This is why I question what type of integrals the hyperphysics page is using.In short, is the hyperphysics page using line integrals, surface integrals, or some other type of integral? If it is using line integrals, then which type of line integral is each equation using. Are Gauss' two laws the type of vector field line integrals that use the curl, while Faraday's law of induction and Ampere's law are the type that uses the divergence (or vice versa)? If you want to know what I mean by these two types, then here is what I mean:
http://tutorial.math.lamar.edu/Classes/CalcIII/CurlDivergence.aspx (last two blue boxes)

 

[Simon Bridge]

Your problem is unclear as you have described it - please be specific.

I'll explain:

I noticed that the notations for those integrals do not match the notations for line integrals of vector fields that I learned ...

It is very common for notation to differ between sources - you get used to it. You need to use context to make the correct associations.

This makes me question whether the integrals in the Maxwell equations are actually even line integrals or if they are surface integrals.

To tell the difference, use the physical situation they are describing and not the notation for clues.
Maxwell's equations are better understood in differential form though.

is the hyperphysics page using line integrals, surface integrals, or some other type of integral?

The hyperphysics page you link to spells out which they mean for each one in the following sections. Each description has a link to an example that also makes the matter clear.

Are Gauss' two laws the type of vector field line integrals that use the curl, while Faraday's law of induction and Ampere's law are the type that uses the divergence (or vice versa)?

... this is spelled out in the second section where the names of the equations are stated followed by the differential form as divergence and curl.
perhaps you will benefit from a refresher in nabla notation: https://en.wikipedia.org/wiki/Del

 

[stedwards]

There is no Maxwell equation in integral form in the second web page.

 

[Simon Bridge]

I understood the second page (the Paul's Calculus notes page) was introduced to show notation learned for integrals of vector spaces.
However, the page only has two integrals in it - div and curl forms of Green's Theorem.

The page does not teach integral notation for vector fields - instead it teaches the basics of div and curl.
The notation in the hyperphysics page is consistent with the notation used in Paul's Maths notes.
Note: $\vec k\mathrm{d}A = \mathrm{d}\vec A = \vec{\mathrm{d} A}$ because Paul uses $\vec k$ as the standard unit normal to the surface element dA while hyperphysics combines the two in the same notation. Perhaps this is where the confusion arises?