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.

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✍️Mathematical Proof

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Vector Properties Dictate the Solution

The final answer of zero isn't a coincidence; it's a direct result of the properties of the vector fields. The fact that $v$ is curl-free and $w$ is divergence-free are the two critical pieces of information that lead to the simplification of the integral.

The Power of Vector Identities

The solution demonstrates the practical utility of vector identities. By using the product rule for divergence, the complex integrand $(\nabla \phi) \cdot w$ was transformed into a simpler form, $\nabla \cdot(\phi w)$, which was then solvable using the Divergence Theorem. Knowing and applying these identities is a key skill in solving vector calculus problems.

The Divergence Theorem as a Problem-Solving Tool

This analysis is an excellent example of the Divergence Theorem's power. It shows how the theorem can be used to convert a difficult volume integral into a simpler surface integral. This transformation is often a crucial step in solving physics and engineering problems.

Boundary Conditions are Crucial

The final result of zero is ultimately determined by the boundary condition. The fact that $w$ is orthogonal to the surface normal at the boundary makes the entire surface integral vanish. This shows that in many problems, the behavior of a field at the boundary is just as important as its behavior within the volume.

🎬Demonstration

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.

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