Divergence and curl are key to understanding vector fields. Divergence measures a field's expansion or compression, with positive values indicating a source, as seen in the demo's outward-flowing vectors. Curl quantifies a field's rotation. The demo's Rotational and Vortex Fields have non-zero curl, showing vectors spinning around a point. The visualization highlights that these properties are independent: a field can have rotation without expansion, and vice-versa. Additionally, a parameter can be used to control the strength and direction of a field, as demonstrated by the Rotational Field.

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✍️Mathematical Proof

$\gg$Mathematical Structures Underlying Physical Laws

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• Divergence and Sources: A vector field with a non-zero divergence is one that expands or compresses. This is clearly shown by the Source Field in option (a), where the positive divergence value of 2 corresponds to vectors radiating outward from the origin.
• Curl and Rotation: A vector field with a non-zero curl is one that exhibits rotation. Both the Rotational Field in option (b) and the Fixed Vortex Field in option (c) have a non-zero curl, which you can see in the demo as vectors spinning around the central point. A zero curl value, as in the Source Field, indicates that the field is irrotational.
• Independent Concepts: The demo highlights that divergence and curl are independent properties. A field can have rotation without expansion (zero divergence, non-zero curl) and expansion without rotation (non-zero divergence, zero curl). This is a fundamental concept in vector calculus.
• Parameterization: The demo for the Rotational Field (b) demonstrates how a constant parameter, in this case a , can be used to control the strength and direction of the rotation, a key aspect of defining vector fields. The Fixed Vortex Field (c) is a specific instance of a rotational field with a constant curl.

🎬Demonstration

Divergence and curl are powerful mathematical tools for describing the behavior of a vector field. Divergence visually corresponds to the expansion or contraction of the field, as seen in the Source Field, while curl corresponds to the rotation of the field, as seen in the Rotational and Vortex Fields. This visualizer helps you see how these abstract concepts relate directly to the physical "flow" represented by the vectors.

Compute the divergence and curl of vector fields

Compute the divergence and curl of vector fields

✍️Mathematical Proof

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Proving the Cross Product Rules with the Levi-Civita Symbol

Proving the Epsilon-Delta Relation and the Bac-Cab Rule

Simplifying Levi-Civita and Kronecker Delta Identities

Dot Cross and Triple Products

Why a Cube's Diagonal Angle Never Changes

How the Cross Product Relates to the Sine of an Angle

Finding the Shortest Distance and Proving Orthogonality for Skew Lines

A Study of Helical Trajectories and Vector Dynamics

The Power of Cross Products: A Visual Guide to Precessing Vectors

Divergence and Curl Analysis of Vector Fields

Unpacking Vector Identities: How to Apply Divergence and Curl Rules

Commutativity and Anti-symmetry in Vector Calculus Identities

Double Curl Identity Proof using the epsilon-delta Relation

The Orthogonality of the Cross Product Proved by the Levi-Civita Symbol and Index Notation

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