# Analysis of a Divergence-Free Vector Field

The analysis of the divergence-free vector field v and its vector potential A visually and mathematically demonstrates a fundamental principle of vector calculus: a field with no sources or sinks can be fully described by a second field-its vector potential-whose curl generates the first field. This shows that the rotational nature of A gives rise to the outward flow of v, linking the two seemingly distinct properties in an elegant and consistent way.

### :clapper:the two vector fields have distinct behaviors with the rotational potential field A acting as the source for the radial field v through the curl operation

> The animation is the visual confirmation of the relationship between a divergence-free vector field and its vector potential. The radial field v shows vectors decreasing in length as they move away from the center. This visualizes the concept of a source without a sink-a field that, while expanding, maintains a constant "flux" across any given surface. The animation's pulsating effect reinforces that the field is present everywhere, but its source is not a single point; rather, the "flux" is conserved. The vector potential $A$ shows vectors that are purely rotational, circling the origin. This swirling pattern is a visual representation of a field whose curl generates the radial field of v. The animation demonstrates that the "twisting" motion of A is what gives rise to the outward "flow" of v. This is a crucial concept, as it shows that a divergence-free field can be described as the curl of another field, its vector potential.

{% embed url="https://youtu.be/lIRrTmHCwN4" %}

### :pen\_ballpoint:Mathematical Proof

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