📢Proving Angular Gradient Identity in 3D

The derivation sheet and demo focus on the integration of abstract mathematical identities and dynamic 3D visualisation to understand vector fields. Users are tasked with proving a complex vector relation involving nested cross-products of position and gradient operators applied to a scalar field. This theoretical work is complemented by an interactive application that tracks a point's movement along a Lissajous curve, allowing for the real-time observation of the relationship between position and gradient vectors. By switching between different mathematical landscapes, such as linear or logarithmic fields, the sources allow for a comparative study of how the direction of steepest functional increase behaves across varying environments.

📎IllustraDemo

Description

This illustration presents an Interactive 3D Vector Field Explorer, a digital tool designed to visualize the relationship between position vectors and gradient vectors within various scalar fields.

Core Visualisation Components

The central feature of the interface is a dynamic 3D plot that demonstrates how vectors interact with a mathematical landscape:

• Dynamic Vector Relationship: The tool visualizes the direct link between a position vector and a gradient vector.

• Color-Coded Vectors: To ensure clarity, a red arrow represents the position vector, while a blue arrow represents the gradient vector, both updating in real-time.

• Lissajous Curve Movement: The point of interest does not move randomly; it traces a pre-defined path through the field known as a Lissajous curve.

Interactive Features and Controls

The tool provides several ways for users to manipulate and analyze the data:

• 3D Camera Control: Users can orbit the scene and zoom in or out to inspect the vectors from any angle.

• Switchable Landscapes: The interface allows users to toggle between different scalar fields, including:

  • Exponential

  • Logarithmic

  • Hyperbolic

  • Linear

• Real-Time Information Panel: A data overlay tracks the underlying mathematical formulas and updates the specific $(x, y, z)$ vector coordinates as the point moves.

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This illustration, titled "Visualising the Geometry of Angular Momentum: From Gradients to Vorticity," provides a geometric bridge between abstract mathematical operators and their physical behaviors. It primary contrasts the Generator of Translation ($\nabla$), which produces linear motion and straight-line vector fields, with the Generator of Rotation ($x \times \nabla$), which defines the orbital angular momentum operator and creates circular, vortex-like paths. The graphic emphasizes the Non-Commutative nature of 3D rotations—noting that the order of operations changes the final outcome—and highlights the Lever Arm Effect, where rotational strength is null at the origin but scales outward with distance. By mapping these operators to specific physical roles like "Linear Force" and "Angular Momentum," the illustration clarifies how symmetry generators dictate the fundamental movement of systems in space.

📎Visualizing Geometry of Angular Momentum

Description

This illustration, titled "Visualising the Geometry of Angular Momentum: From Gradients to Vorticity," provides a conceptual and geometric breakdown of how mathematical operators translate into physical motion.

Core Operator Comparison

The graphic categorizes two fundamental operators through their physical roles and geometric results:

• The Generator of Translation ($\nabla$): Represented as the gradient, this operator acts as a straight-line vector field. Physically, it corresponds to a linear force, and its geometric result is translation (moving an object from one point to another).

• The Generator of Rotation ($x \times \nabla$): This operator defines Orbital Angular Momentum. It transforms linear vectors into circular paths, resulting in a rotation or vortex geometry.

Key Physical Insights

The illustration highlights three critical properties of these fields:

• Radial Symmetry (Translation): The left visual shows a starburst pattern where vectors point directly toward or away from a central point, illustrating the gradient's linear nature.

• Non-Commutative Flow: The central and right visuals demonstrate that the order of operations matters in 3D space. Specifically, rotating around the X-axis then the Y-axis produces a different outcome than the reverse order.

• The Lever Arm Effect: The graphic explains that rotational strength is zero at the origin and increases as the distance from the center increases, reflecting the $x$ component of the angular momentum operator.

Visual Summary Table

Operator	Physical Role	Geometric Result
$\nabla$ (Gradient)	Linear Force	Translation
$x \times \nabla$	Angular Momentum	Rotation/Vortex

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🧵Related Derivation

🧄Proof and Implications of a Vector Operator Identity (VOI)

⚒️Compound Page

Proving Angular Gradient Identity in 3D | IllustraDemoviadean on Notion