I am trying to understand the basic set up - the problem - behind line integration of vector fields, and I found this great applet written by Matthias Kawski of the Mathematics Department at Arizona State University online, which after much trouble installing, allowed me to follow this lecture.
The applet shows the vector field given by $\tiny\begin{bmatrix}x^2\\y^2\end{bmatrix}$ as:
The applet allows the user to produce the stacks, orthogonal to the vector field arrows, corresponding to the differential 1-forms (covariant vectors):
and to draw an integration line manually as it calculates the value of the line integral as circ on the right side of the panel:
which amounts to $18.463980$ for this particular line. As the stacks are pierced they are nicely highlighted in green, giving a tactile intuition of what the integral is doing.
The question is about the Flux value also in the right panel, and in this case corresponding to $-1.462592.$ This is not so clearly defined in the linked instruction of the applet.
If I can imagine the flux as water (or electricity) running through an infinitesimally small parallelogram on a surface with the velocity of flow corresponding to the vector field $\tiny\begin{bmatrix}x^2\\y^2\end{bmatrix}$, I don't see where the surface in question is being defined. On the other hand, $\tiny\begin{bmatrix}x^2\\y^2\end{bmatrix}$ can depict a surface, but then the vector field describing the velocity of water (or electricity) flowing through this surface is not defined.
I don't care about the applet per se, but in trying to understand what it is possibly being calculated, I attempt to see the set up for integration along a vector field.
EDIT after comments: There is no "surface" in this example: just a curve (in red, below), instead, with the vector field indicating the velocity of some fluid flowing at each point on $\mathbb R^2$ (circ). The "flux" quantifies the flow orthogonal to the vector field:
Here is an example:
