🧄Integral of a Curl-Free Vector Field

The integral of a curl-free field dotted with a divergence-free field over a closed volume is zero when the divergence-free field is tangential to the boundary, a result that highlights the critical role of vector identities, the Divergence Theorem, and boundary conditions.

🎬the integral of a vector field with zero curl and a vector field with zero divergence over a closed volume is zero

the integral of a vector field with zero curl ( $v$ ) and a vector field with zero divergence ( $w$ ) over a closed volume is zero, provided that $w$ is tangential to the boundary. This is a visual demonstration of the relationship between different types of vector fields and their properties within a defined volume. The integrand, which represents the dot product of the two fields $(v \cdot w)$, has both positive (red) and negative (blue) contributions that cancel each other out due to the symmetry of the fields.

🖊️Mathematical Proof

Integral of a Curl-Free Vector Field | Proof and Derivationviadean on Notion