## Boundaries and Flows

Gauß’ divergence theorem states that, unless matter is created or destroyed, the density within a region of space V can change only by flowing through its boundary ∂V. Therefore

$\large \dpi{150} \bg_white \int_{\partial V} \mathbf{flow}\ d \, \text{surface area} \quad = \quad \int_V \nabla \mathbf{flow}\ d \, \text{volume}$

i.e., you can measure the changes in an entire region by simply measuring what passes in and out of the boundaries of the region.

"Stuff passing through a boundary " could be:

• tigers through a conservation zone (2-D)
• sodium ions through a biological cell (3-D)

• magnetic flux through a toroidal fusion chamber

• water through a reservoir (but you’d have to measure evaporation, rain, dew/condensation, and ground seepage in order to get all of ∂V)

• in the other direction, you could measure water upstream and downstream in a river (no tributaries in between) and infer the net amount of water that was drunk, evaporated, or seeped

• probability mass through a set of possibilities

• particulate pollution through "greater Los Angeles"

• ¿ notes through a symphonic orchestra ?
• chromium(VI) through a human body
• smoke or steam through an industrial cooling tower or smokestack

• imports and exports through an economy

• goods or cash through a limited liability company

Said in words, the observation that you can measure change within an entire region by just measuring all of its boundaries sounds obvious, even trivial. Said symbolically, Gauß’ discovery amounts to a nifty tradeoff between boundaries  and gradients . (The gradient  is the net amount of a flow: flow in direction 1 plus flow in orthogonal direction 2 plus flow in mutually orthogonal direction 3 plus…) It also amounts to a connection between 2-D and 3-D.



Because of Cartan-style differential geometry, we know that the connection is much more general: 1-D shapes bound 2-D shapes, 77-D shapes bound 78-D shapes, and so on.

Nice one, Fred.

(Source: ocw.mit.edu)

45 notes

1. paolosdala reblogged this from proofmathisbeautiful
2. sentimentalrobot reblogged this from proofmathisbeautiful and added:
I remember being asked to explain this and other to my peers on a few days before the final. Felt really cool knowing...
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7. martinstheorem said: Shouldn’t there be a dot product between the flow and surface element on the left and the divergence of the flow on the right? Unless you’re using some obscure notation.
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