# 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.

### :clapper: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.

{% embed url="https://youtu.be/Q-2TAugvPlc" %}

### :pen\_ballpoint:Mathematical Proof

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