This proof is that the integral vanishes because the integrand can be rewritten as the curl of a vector field $(\phi \nabla \psi)$. This allows the application of Stokes' Theorem, shifting the focus from the entire surface $S$ to its boundary $C$. Since $\phi$ is constant on that boundary, it acts as a uniform scaling factor that can be moved outside the integral, leaving only the circulation of a gradient field $(\nabla \psi)$ around a closed loop. Because gradient fields are conservative, their path integral around any closed loop is identically zero, regardless of the complexity of the surface or the specific nature of the scalar fields involved.
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title Geometric Equilibrium: Mathematical Proofs and Physical Visualisations of Stokes' Theorem
Resulmation: Surface integral Proof
: Conservative Force - Work Around a Closed Loop
: Surface Integral Proof via Stokes' Theorem
IllustraDemo: Constant Boundaries Cancel Surface Integrals
Ex-Demo: The Geometry of Equilibrium and Conservative Forces
Narr-graphic: The Geometry of Conservative Forces and Stokes' Theorem
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