🧄Unpacking Vector Identities: How to Apply Divergence and Curl Rules (VI-DCR)

These derivations serve as a powerful illustration of applying vector calculus identities, particularly leveraging the simple, well-known properties of the position vector $x$ , specifically that its divergence is a constant (3) and its curl is zero. The key takeaways confirm the structure of fundamental identities: for instance, the divergence of the cross product $\nabla \cdot(x \times \nabla \phi$ ) vanishes completely because both $x$ and any gradient field ( $\nabla \phi$ ) are irrotational. Conversely, expanding the divergence of the product $\nabla \cdot(\phi \nabla \phi)$ naturally produced the two crucial components for characterizing a scalar field's variation: the Laplacian $(\phi \Delta \phi)$ and the squared magnitude of the gradient $\left(|\nabla \phi|^2\right)$ , demonstrating how basic differential operations often lead back to the most important second-order field equations.