Calculus
Gauss's Divergence Theorem
Fill a lumpy region with little boxes and ask each one how much more flow leaves it than enters it. Every inner wall is shared by two boxes, so whatever leaves one enters the other and the pair cancels. Add every box and only the outer skin is left.
flow leavingflow enteringthe box you picked
Click any box to open it up · in the solid view, drag to orbit
Box you picked—
—
View
Field
a few big oneshundreds of tiny ones
flow out through the skin—
leak added over the boxes—
gap between the two—
What to observe
- Click a box. The card on the right puts an arrow through each wall,red where flow pours out and blue where it pours in. Out minus in, divided by the box's size, is its divergence.
- Tick cancel the shared walls. Whatever leaves one box enters its neighbour, so the inner walls dissolve in pairs and only the outer skin is left. That is the theorem.
- On Hot spot all the divergence sits in one lump in the middle. Only the boxes over the lump light up, yet the skin far away reports all of it.
- On Swirl the flow is busy but nothing accumulates: every box is balanced and the skin reports zero. Spinning is not spreading.
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