🧄Divergence and Curl Analysis of Vector Fields (DCA-VF)

Here is the relationship between the geometric structure of a vector field and its differential operators. The position vector $x$ is purely radial and expanding, resulting in a constant positive divergence but no curl. In contrast, fields like $a \times x$ and $v_2$ describe rotational motion; they are "solenoidal" (meaning their divergence is zero), but they possess constant "vorticity" or curl. These examples demonstrate that divergence measures the density of "sources" or "sinks" at a point, while curl quantifies the local circulation or rotation around that point.

🎬Narrated Video

• Demo

🎬Visualization of Three Vector Fields with Different Divergence and Curl (TVF-DC)

📎IllustraDemo

• Illustration

📢Divergence Measures Flow Curl Measures Spin (DF-CS)

🧣Example-to-Demo

• Flowchart and Mindmap

🧣Divergence and Curl of Vector Fields (DC-VF)

🍁A Visual and Mathematical Synthesis of Vector Fields

Description

These documents map the relationship between mathematical theory, digital demonstration, and visual intuition for divergence and curl in vector fields. The flowchart outlines a technical workflow that bridges calculus definitions with Python and HTML-based simulations to categorize fields like "Source" or "Fixed Vortex" based on their flow characteristics. The mindmap streamlines these concepts into a logical hierarchy, defining divergence as flow behavior (expansion or compression) and curl as rotational tendency, while providing specific mathematical results for position, cross product, and planar rotational fields. Finally, the illustration offers a high-level visual comparison, contrasting the radiating "source" behavior of the position vector (non-zero divergence) with the circulating "solenoidal" behavior of circulation vectors (non-zero curl) to ground abstract math in physical movement.

Key points

• Flowchart Utility: Details the progression from mathematical definitions to interactive digital demos, ultimately identifying physical flow properties.

• Mindmap Structure: Organizes divergence and curl into "Key Concepts" (definitions) and "Mathematical Analysis" (calculated results for specific fields).

• Visual Illustration: Physically depicts divergence as outward flow from a point and curl as local vorticity or circulation.

• Divergence Logic: Measures a field's tendency to act as a source or sink; zero divergence implies a solenoidal field.

• Curl Logic: Measures local rotation; zero curl defines an irrotational field.

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⚒️Compound Page

Divergence and Curl Analysis of Vector Fields | Proof and Derivationviadean on Notion