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

This visualization comprehensively demonstrates how the flux integral of the vector field $v=\left(x^k, y^k, z^k\right)$ through a sphere is determined by the parity of the exponent $k$. The animation clearly shows that when $k$ is odd (e.g., $k=1,3$ ), the field is directed consistently outward, resulting in a positive local flux density across the entire surface and a calculated positive total flux $\Phi$ ; conversely, when $k$ is even (e.g., $k= 2,4$ ), the field components remain positive, causing the vector field to exhibit symmetric inward and outward flow patterns (indicated by balanced regions of positive and negative local flux density), which precisely cancel one another across the spherically symmetric domain, confirming the theoretical zero total flux ( $\Phi=0$ ) predicted by the Divergence Theorem.

🎬Narrated Video

The "Translational Variance of Divergent Fields" demo illustrates that the flux of a vector field is invariant under translation only when the field's divergence is constant (the k=1 case). For higher-order fields (k>1), the divergence is spatially dependent, meaning it varies throughout the coordinate system. When the sphere is centered at the origin, the geometric symmetry perfectly cancels out the flux for even values of $k$ because the divergence acts as an odd function over a symmetric domain. Shifting the sphere breaks this symmetry. As the sphere moves into regions of higher field intensity, the opposing contributions no longer balance out over the volume. This results in a flux that is a deterministic polynomial function of the sphere's position (a, b, c), demonstrating that in non-uniform fields, the net flow through a surface is determined by its specific location relative to the field's gradients.