🧄Unpacking Vector Identities: How to Apply Divergence and Curl Rules

Vector calculus product rules systematically simplify complex expressions by breaking them into fundamental components. Essential identities like $\nabla \cdot x=3$ and $\nabla \times(\nabla \phi)=0$ are critical for this process, often causing terms to vanish. The visualizations powerfully demonstrate that the final vector field is a superposition of these simpler effects. For example, the animations show that the divergence of a product field is a sum of distinct terms, while a cross-product divergence can resolve to zero because its constituent parts do.

🎬Divergence Product Rule Visualization

the divergence of a cross product rule resulting in zero everywhere

the divergence of a scalar field times its own gradient

the use of the BAC-CAB rule for the curl of a cross product

🧄Mathematical Proof

Unpacking Vector Identities: How to Apply Divergence and Curl Rules | Proof and Derivationviadean on Notion