Unpacking Vector Identities: How to Apply Divergence and Curl Rules

Downloadable Files

• Derivation Sheet.md
• Code Snippets.rar
• Code Snippets with Diagrams.md

Summary

These files describe a series of four Python-based 3D animations that visualize fundamental vector calculus identities using a Gaussian scalar field. These demonstrations progressively build from basic divergence properties of position-scaled fields and null identities—where results such as the curl of a gradient are identically zero—to more complex second-order operations like Green's First Identity and the curl of a cross product. Each script follows a consistent logical structure: defining analytical derivatives (gradient, Laplacian, and directional derivatives), computing mathematical terms across a 3D meshgrid, and utilizing matplotlib to render either scatter plots for magnitude or quiver plots for directional flow. By decomposing these complex identities into individual components through FuncAnimation cycles, the project illustrates the geometric and physical consequences of differential operators, showing how distinct vector fields superimpose to model phenomena like transport and rotation.

Kanban

Kanban:Field Architectures: The Visual Logic of Differential Identities