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

https://youtu.be/ECYdzO22sCI

🏗️Structural clarification of Poof and Derivation

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🗒️Downloadable Files - Recursive updates (Feb 10,2026)

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All content and images on this page are the property of Sayako Dean, unless otherwise stated. They are protected by United States and international copyright laws. Any unauthorized use, reproduction, or distribution is strictly prohibited. For permission requests, please contact [email protected]

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