Divergence and Curl Analysis of Vector Fields

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

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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