🧄Conditions for a Scalar Field Identity (SFI)

The identity holds if and only if the scalar field $\phi$ is harmonic. By applying vector expansion identities and leveraging the fact that $a$ is constant, the complex directional derivatives on both sides of the equation cancel out. This leaves the expression dependent solely on the product of the vector $a$ and the Laplacian of the scalar field, $\nabla^2 \phi$. Since $a$ is non-zero, the relation forces the Laplacian to vanish, meaning $\phi$ must satisfy Laplace's Equation. This result highlights how the curl of a cross product involving a gradient simplifies significantly when one component is a constant field, ultimately linking the vector identity to the fundamental properties of potential theory.

🎬Narrated Video

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