The visualizations of complex vector identities reveal the geometric consequences of differential operators, particularly leveraging the simple properties of the position vector. The divergence product rule $\nabla \cdot(\phi \vec{x})$ demonstrates that the total outflow is an additive superposition: the position vector's intrinsic divergence is scaled by the scalar field, plus an effect driven by the field's gradient. Crucially, the animation for $\nabla \cdot(\vec{x} \times \nabla \phi)$ provides a visual proof that the divergence of this cross product is identically zero, as both component terms vanish. Analyzing $\nabla \cdot(\phi \nabla \phi)$ highlights its fundamental decomposition into the squared gradient magnitude and the product of the scalar field and its Laplacian $(\Delta \phi)$, a result essential for modeling transport phenomena. Finally, the visualization of the complex curl identity $\nabla \times(\vec{x} \times \nabla \phi)$ using the BAC-CAB rule shows how multiple distinct vector fields combine to define the intricate resulting rotation.
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%% Proof and Derivation
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