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

The total flux ( Φ\Phi ) of a vector field through a closed surface is critically determined by the parity of the integer kk in the vector field's definition. If kk 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 kk 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 Φ=12πRk+2k+2\Phi=\frac{12 \pi R^{k+2}}{k+2}. This illustrates how the nature of the vector field, influenced by kk, 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

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