🧄Circulation Integral vs. Surface Integral

The circulation integral of the position vector around a closed loop equals twice the vector area of the enclosed surface. The line integral of the cross product of the position vector and differential path vector is equivalent to twice the vector area of the surface enclosed by the loop. This allows for the simplification of complex calculations into direct geometric measurements. The result can be verified by checking the ratio of the calculated integral to the vector area, which converges to 2 as the approximation improves, providing a crucial verification of the result through the agreement between direct integration and the use of the theorem.

🎬A discrete sum converges on the true value of a continuous integral

The line integral of the cross product of the position vector and the differential path vector is not zero. It's equivalent to twice the vector area of the surface enclosed by the loop. This means a complex calculation can be simplified into a direct geometric measurement, and the result can be verified by observing the ratio of the calculated integral to the vector area converge to 2 as the approximation improves.

A discrete sum converges on the true value of a continuous integral

🖊️Mathematical Proof

Circulation Integral vs. Surface Integral | Proof and Derivationviadean on Notion