🎬Stokes' Theorem in 3D-Comparing Geometric Area to General Circulation on a Saddle Loop

This comprehensive demonstration of Stokes' Theorem spanned both analytical physics and numerical methods, starting by establishing the geometric relationship that the integral $I= \oint_{ T } x \times d x$ simplifies to exactly twice the vector area $\left(I=2 \iint_S d S\right)$, a conclusion reinforced by applying it to a non-planar saddle loop where symmetry dictated that the result depended only on the flat $x y$-projection. The analysis then shifted to a general vector field, $A=$ ( $z^2, x^2, y^2$ ), demonstrating that the integral depends on the complex, spatially varying interaction between the non-constant curl, $\nabla \times A$, and the local surface orientation, resulting in a specific scalar value of $I=-\frac{\pi r_0^4}{2}$. Finally, the interactive demo provided practical verification of the geometric principle by illustrating numerical convergence, showing that as the continuous loop was approximated by an increasing number of discrete polygon segments, the calculated integral's ratio to the theoretical vector area $(I / A)$ steadily converged toward the ideal value of 2, thus bridging the gap between abstract vector calculus and practical computational physics.

🎬Narrated Video

🧵Related Derivation

🧄Circulation Integral vs. Surface Integral (CI-SI)

⚒️Compound Page

Stokes' Theorem in 3D-Comparing Geometric Area to General Circulation on a Saddle Loop | Resulmationviadean on Notion