I just made a physical model that is easy to understand by Green's formula. Please forgive me if it is not rigorous, but it should be quite rigorous.
Physical Model of a Wind Turbine
It is assumed that a large fan is used to generate power (the picture of the wind power station is not used here). The shape of the fan is shown in the figure below:
for the sake of understanding and drawing, the fan is simplified:
The energy obtained by the fan
The center of the fan is understood as a "singularity", and the two blue curves form a surface D; also using the idea of differentiation, take a point on the fan for analysis, and the point moves in a circular motion in the horizontal direction. The work is the product of the force F received in the horizontal direction and the moving path (velocity v×time t), as shown in the figure below; the
form converted into calculus is the work done by the force along the circumferential direction.
work done by the wind
Also consider that the wind pressure on this point is P, and the area of this point is S, then the force of the wind force acting on this point is PS, and the work done by the wind force is the product of the force in the vertical direction and the wind displacement (wind speed v×time t) . The magnitude of the work done by the wind force is:
the form transformed into calculus is the double integral of the product of pressure and wind displacement on the area.
Energy conservation law
According to the law of energy conservation, the two are equal, the first part is understood as the circulation, and the second part is understood as the flux, that is to say, the flux enclosed by the closed surface and the circulation of the curve enclosed by the surface are determined relational .
Observe carefully that this is what the Gaussian formula says. The left side is the surface integral (flux), and the right side is the curve integral (circumference), and the two are equal.
Maxwell's equations
Among them, Faraday's law of electromagnetic induction of Maxwell's equation is this formula. Have time to expand on it.