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.

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✍️Mathematical Proof

$\gg$Mathematical Structures Underlying Physical Laws

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The most critical factor in determining the flux integral $\Phi$ is whether the integer $k$ is even or odd. This is because the parity of $k$ dictates the fundamental behavior of the vector field $v$.

The Case of Even $k$

When $k$ is an even positive integer, the components of the vector field (e.g., $\left(x^1\right)^k$ ) are always positive, regardless of the coordinate's sign. This creates a highly symmetrical vector field that points into certain octants. As a result, the flow of the vector field entering the sphere on one side is perfectly balanced by the flow exiting on the other, leading to a net flux of zero. This outcome is a powerful illustration of physical symmetry causing a cancellation of effects.

The Case of Odd $k$

When $k$ is an odd positive integer, the components of the vector field (e.g., $\left(x^1\right)^k$ ) retain the sign of their respective coordinates. This means the vector field is perfectly radial, with every vector pointing directly away from the origin. Because the vector field is perfectly aligned with the outward-pointing surface normal of the sphere, the total flow is exclusively outward, resulting in a positive, non-zero flux. The exact value is given by the formula $\Phi=\frac{12 \pi R^{k+2}}{k+2}$.

🎬Demonstration

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