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.

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✍️Mathematical Proof

$\gg$Mathematical Structures Underlying Physical Laws

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Understanding Product Rules is Essential: The core of the problem is the correct application of three key vector calculus product rules: the divergence of a scalar times a vector, the divergence of a cross product, and the curl of a cross product (BAC-CAB rule). Each expression is rewritten by systematically applying these rules.
Key Identities Simplify Complex Expressions: Two fundamental identities of the position vector, $\nabla \cdot x=3$ and $\nabla \times x=0$, are crucial for simplifying the expressions. For example, in part (b), these identities are what cause the entire expression to resolve to zero.
The Curl of a Gradient is Always Zero: A critical identity used in part (b) is that $\nabla \times$ $(\nabla \phi)=0$. This means that the curl of any conservative vector field (one that can be expressed as a gradient) is always zero, a concept central to physics and engineering.
The Result of $\nabla \cdot(\phi \nabla \phi)$ : The expansion of this specific expression reveals a connection to the Laplacian, $\nabla^2 \phi$. The simplified result, $|\nabla \phi|^2+\phi \nabla^2 \phi$, is a common and important form in fields like partial differential equations, particularly the diffusion equation.
Systematic Simplification: The analysis of part (d) demonstrates that even very complicated vector operations, like the curl of a cross product, can be broken down and simplified step-by-step using known product rules and vector identities, revealing the underlying structure of the expression.

🎬Demonstration

The animation for this product rule demonstrates that the total divergence of the combined field is a superposition of two distinct effects. You can visually see how the outward "flow" of the position vector field is amplified by the scalar field (ϕ), and how the divergence of the final field is the sum of the scalar-scaled original divergence and a new term related to the gradient of the scalar field.

Divergence Product Rule Visualization

This animation provides a powerful visual proof that the divergence of this specific cross product is always zero. By animating each term of the product rule, it shows that both terms (the curl of the position vector and the curl of the gradient) individually vanish, confirming that the entire expression is identically zero.

the divergence of a cross product rule resulting in zero everywhere

The decomposition of a complex divergence into two physically meaningful components. The animation visualizes how the final divergence is the sum of the squared magnitude of the gradient (representing the rate of change) and the scalar field multiplied by its Laplacian (related to concentration). This decomposition is fundamental to understanding processes like diffusion.

the divergence of a scalar field times its own gradient

This animation demonstrates the complexity and utility of the BAC-CAB rule. It shows that the curl of this cross product is not simply zero. Instead, it is the vector sum of three distinct vector fields, each with its own structure. The visualization allows you to see how these individual vector fields combine to produce the final, more intricate vector field that represents the curl of the original expression.

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

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