how the Divergence Theorem proves the formulas for the volume and surface area of a sphere

This interactive demonstration uses a variety of features to help users understand the Divergence Theorem. In a 3D view, you can see a transparent sphere with a position vector and a surface area vector, which you can manipulate by orbiting and zooming. The 2D view shows a circle, where 'volume' is its area and 'surface area' is its circumference, and you can change its radius by dragging it. A radius slider provides precise control. An info panel updates in real-time with the theoretical and calculated values, showing how the volume (or area) derived from the theorem, $V=\frac{A R}{N}$, always matches the actual calculated volume, thereby confirming the theorem's validity in both 2 D and 3 D .

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