Meaning of a scalar surface integral of a vector field?

Question

I apologize if the title is confusing but after going through some assignments I was able to correctly derive that $\iiint_{V}(\overrightarrow{\nabla}\times A)dV=-\oint_{\partial V} (A \times \hat{n})dS$. Where the right hand side should be a closed surface integral but unfortunately I don't know a lot of latex. In this case we have that $A$ is a vector field and in the derivation I had that $d\textbf{S}=\hat{n}dS$ so in this case our vector integral became a scalar integral but it is a scalar surface integral of a vector field. What exactly is the meaning of this?

Answer 1

It looks like you are investigating Stokes' Theorem.

Let's start with a line integral. When I think of the line integral, I imagine that I am working against a force. Sometimes it is in my face and I must spend energy to fight it, and sometimes it is at my back, and I extract energy from the forcefield (like riding a bicycle downhill). If you are working in a conservative force field then the energy you get out in the force-at-your-back segments equals what you spend in the fighting-the-teeth-of-the-force sections. Gravity is a classic example of a conservative force.

If your force field is non-conservative, then there is a net energy that you must put in (or take out) to complete a circuit that returns you to your starting point. I visualize a spiraling action to the vectors of the force field. You are fighting a cyclone that causes you to work against the force the whole way around the circuit. In reality, there may be sections when the wind is at your back and times when it is in your face, but there is a net rotation to the system.

When you take the curl of the vector field, you are capturing an expression of how much spinning action there is at each point in the force field. You can sum the rotating action at each point on the surface and it will tell you the total rotation around the boundary.

It is part of a larger set of theorems -- the generalized Stokes theorem -- that say that you can measure the derivative of a function over a manifold and that will equal the expression of the function over the boundary of the manifold.

The fundamental theorem of calculus was your first exposure to the generalized Stokes' theorem.

$f(b)-f(a) = \int_a^b \frac {d}{dt} f(t)\ dt$

You might think about what you are learning now an extension of the FTC to more complicated structures in higher dimensions.