🧄Divergence Theorem Analysis of a Vector Field with Power-Law Components

The total flux ( $\Phi$ ) of a vector field through a closed surface is critically determined by the parity of the integer $k$ in the vector field's definition. If $k$ is even, the vector field's components are always positive, resulting in a symmetrical field where inward and outward flows cancel each other out, leading to zero net flux. If $k$ is odd, the vector field is perfectly radial, with vectors pointing directly away from the origin, resulting in a positive, non-zero flux quantified by $\Phi=\frac{12 \pi R^{k+2}}{k+2}$. This illustrates how the nature of the vector field, influenced by $k$, dictates the net flow across the surface.

🎬Compute the flux integral against the sphere of radius with the surface normal pointing away from the origin

When k is an even integer, the vector field's components are always positive, regardless of the coordinates. This creates a symmetrical pattern where the flow into the sphere on one side is canceled by the flow out on the other, resulting in a zero flux. When k is an odd integer, the vector field's components retain the sign of the coordinates, causing the vectors to point radially outward from the origin. This consistent outward flow leads to a positive flux.

Compute the flux integral against the sphere of radius with the surface normal pointing away from the origin

🖊️Mathematical Proof

Divergence Theorem Analysis of a Vector Field with Power-Law Components | Proof and Derivationviadean on Notion