🧄Boundary-Driven Cancellation in Vector Field Integrals

The integral of the dot product of a curl-free field and a divergence-free field over a closed volume is zero. This result is not a coincidence; it's a fundamental principle of vector calculus, visually demonstrated by the orthogonality of the two vector fields. The visualization highlights how this can be proven mathematically using vector identities and the Divergence Theorem, which converts a difficult volume integral into a much simpler surface integral. The final result of zero is then confirmed by the boundary condition that the divergence-free field is tangential to the surface of the sphere.

🎬A curl-free field and a divergence-free field within a translucent sphere

The integral of the dot product of a curl-free field and a divergence-free field over a volume is zero if both fields are well-behaved. The visualization demonstrates this by showing how the two types of vector fields are orthogonal (at a 90-degree angle) to each other everywhere within the sphere. Because the dot product of two orthogonal vectors is always zero, the total integral over the entire volume also evaluates to zero. This is a fundamental concept in vector calculus and physics.

🖊️Mathematical Proof

Boundary-Driven Cancellation in Vector Field Integrals | Proof and Derivationviadean on Notion