📢Divergence Curl and Diffusion Identities

The sources focus on the simplification of complex vector identities by expressing them solely in terms of a scalar field $\phi$ and its derivatives, thereby removing the need for derivatives of the position vector $\vec{x}$ . This mathematical decomposition reveals that differential operators have significant geometric and physical consequences, such as showing that the divergence product rule acts as an additive superposition of the field's gradient and the position vector's intrinsic divergence. Notably, certain identities like $\nabla \cdot(\vec{x} \times \nabla \phi)$ are shown to be identically zero, while the expansion of $\nabla \cdot(\phi \nabla \phi)$ into the squared gradient magnitude and the Laplacian ( $\Delta \phi$ ) is identified as a fundamental requirement for modelling transport phenomena. Finally, the sources highlight that complex rotations, such as those found in curl identities, can be understood by combining distinct vector fields using tools like the BAC-CAB rule.

📎IllustraDemo

Description

This illustration, titled "Vector Calculus Identities Visualised," provides a graphical representation of how complex vector expressions decompose into intuitive components used for modeling physical phenomena like transport and rotation.

The visualization breaks down four key identities involving a scalar field ($\phi$) and a position vector ( $\vec{x}$ ):

1. Identity 1: $\nabla \cdot (\phi \vec{x})$

Concept: This identity represents a superposition of the field's gradient and its magnitude.
Visual Breakdown: The total outflow combines the gradient of the field ( $\nabla \phi$ ) with its intrinsic divergence scaling. The mindmap identifies these components as $x \cdot \nabla \phi$ and $3\phi$ .

2. Identity 2: $\nabla \cdot (\vec{x} \times \nabla \phi)$

Concept: The divergence of this specific cross product is always zero.
Visual Breakdown: The illustration shows a "null" symbol ( $\emptyset$ ), providing a visual proof that all component terms in this identity vanish completely. This relies on the principle that the curl of a gradient is zero.

3. Identity 3: $\nabla \cdot (\phi \nabla \phi)$

Concept: This expression splits into the gradient magnitude and the Laplacian effect.
Visual Breakdown: The graphic shows a flow splitting into two distinct paths: $|\nabla \phi|^2$ and $\phi \nabla^2 \phi$ . This decomposition is noted as essential for modeling transport phenomena in physics.

4. Identity 4: $\nabla \times (\vec{x} \times \nabla \phi)$

Concept: This curl expression defines a complex resulting rotation.
Visual Breakdown: The illustration uses swirling patterns to show how multiple distinct vector fields combine to create the outcome. According to the mindmap, this involves the BAC-CAB rule and decomposes into three components: $x \nabla^2 \phi$ , $-2\nabla \phi$ , and $-(x \cdot \nabla)(\nabla \phi)$ .