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# Gauss' Theorem

Module by: Paul Padley. E-mail the author

Summary: A simple exposition of Gauss's theorem or the divergence theorem.

## Gauss' Theorem

Consider the following volume enclosed by a surface we will call S S .

Now we will embed S S in a vector field:

We will cut the the object into two volumes that are enclosed by surfaces we will call S 1 S 1 and S 2 S 2 .

Again we embed it in the same vector field. It is clear that flux through S 1 S 1 + S 2 S 2 is equal to flux through S . S . This is because the flux through one side of the plane is exactly opposite to the flux through the other side of the plane: So we see that S F d a = S 1 F d a 1 + S 2 F d a 2 . S F d a = S 1 F d a 1 + S 2 F d a 2 . We could subdivide the surface as much as we want and so for n n subdivisions the integral becomes:

S F d a = i = 1 n S i F d a i . S F d a = i = 1 n S i F d a i . What is S i F d a i S i F d a i .? We can subdivide the volume into a bunch of little cubes:

To first order (which is all that matters since we will take the limit of a small volume) the field at a point at the bottom of the box is F z + Δ x 2 F z x + Δ y 2 F z y F z + Δ x 2 F z x + Δ y 2 F z y where we have assumed the middle of the bottom of the box is the point ( x + Δ x 2 , y + Δ y 2 , z ) ( x + Δ x 2 , y + Δ y 2 , z ) . Through the top of the box ( x + Δ x 2 , y + Δ y 2 , z + Δ z ) ( x + Δ x 2 , y + Δ y 2 , z + Δ z ) you get F z + Δ x 2 F z x + Δ y 2 F z y + Δ z F z z F z + Δ x 2 F z x + Δ y 2 F z y + Δ z F z z Through the top and bottom surfaces you get Flux Top - Flux bottom

Which is Δ x Δ y Δ z F z z = Δ V F z z Δ x Δ y Δ z F z z = Δ V F z z

Likewise you get the same result in the other dimensionsHence S i F d a i = Δ V i [ F x x + F y y + F z z ] S i F d a i = Δ V i [ F x x + F y y + F z z ]

or S i F d a i = F Δ V i S i F d a i = F Δ V i S F d a = i = 1 n S i F d a i = i = 1 n F Δ V i S F d a = i = 1 n S i F d a i = i = 1 n F Δ V i

So in the limit that Δ V i 0 Δ V i 0 and n n S F d a = V F V S F d a = V F V

This result is intimately connected to the fundamental definition of the divergence which is F lim V 0 1 V S F d a F lim V 0 1 V S F d a where the integral is taken over the surface enclosing the volume V V . The divergence is the flux out of a volume, per unit volume, in the limit of an infinitely small volume. By our proof of Gauss' theorem, we have shown that the del operator acting on a vector field captures this definition.

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