I've been reading about the two-dimensional flux and the flux integral, which had not been taught to us, in multivariable calculus. We were only taught the circulation line integral, so I'm quite unfamiliar with this concept.
However, what I came across was the following :
$\oint_\gamma \vec{F}.\hat{n}\space ds$ is the flux of the vector field through the curve.
Using Green's theorem, I can write the above as :
$\oint_\gamma \vec{F}.\hat{n}\space ds = \oint \vec{\nabla}.\vec{F} \space dS $
Hence, I've related the line integral to this surface integral, where the surface is just the surface enclosed by the curve $\gamma$.
This is easy for a curve on the $x-y$ plane, as $dS=dxdy$ and the divergence can be easily found out.
How do we extend this to three dimensions? As in, we are still calculating the two-dimensional flux, but the curve is given by $f(x,y,z)=c$, for example. We cannot use $dS=dxdy$ $\space$ anymore, as we now have some $z$ component of area.
Do I just find out the divergence and then treat this as a regular scalar surface integral and solve this? In that case, we have :
$ds = \sqrt{{f_x}^2 + {f_y}^2 + 1}\space dxdy$
Or, we could parameterize the surface and the curve, and then simply solve the integral in this parametric form, using the parametric variables $u,v$ for example.
Am I correct, or is there something that I'm missing ?
You are reaching toward the "Generalized Stokes Theorem." $\int_d\Omega \omega = \int_\Omega \ d\omega$
So, what does that expression mean? The integral of a differential form over the boundary of some manifold equals the integration of the exterior derivative of said form over the entire manifold.
There are a lot of fancy words in there such as "manfold" and "differential form" and "exterior derivative", that I am not going to define here, or at least not with any rigor or precision. A manifold could be an interval, or a surface or a volume. Lets think of a differential form as a function. And the exterior derivative as the appropriate differential operator for what we want to integrate.
One application you already know is the fundamental theorem of calculus.
$F(b) - F(a) = \int_a^b f(x)\ dx$ where $\frac {d}{dx} F(x) = f(x)$
$F(x),$ and $f(x)$ are differential forms, $[a,b]$ is a manifold.
Greens theorem is also an example.
So, what are the other cases in $\mathbb R^3$ ?
Stokes theorem: If $S$ is a surface embedded in $\mathbb R^3$ and $r$ is the boundary of this surface.
$\oint_r f\cdot dr = \iint \nabla \times f \ dS$
The contour integral around a closed path equals the surface integral of the curl of the function. This reduces to Greens theorem if the surface lies in a plane.
Gauss' Theorem: If $V$ is a volume in $\mathbb R^3$, and $S$ is its surface.
$\unicode{x222F}_S f\ dS = \iiint \nabla \cdot f \ dV$
The flux across the surface equals the integral of the divergence of the function over its volume.