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Gauss' Theorem

Module by: Paul Padley

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

Note: Your browser may not currently support MathML. See our browser support page for additional details. You can always view the correct math in the PDF version.

Gauss' Theorem

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

Figure 1
Figure 1 (Gauss_Law_Drawings_4.gif)
Now we will embed S S in a vector field:
Figure 2
Figure 2 (Gauss_Law_Drawings_6.gif)

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 .

Figure 3
Figure 3 (Gauss_Law_Drawings_5.gif)
Again we embed it in the same vector field.
Figure 4
Figure 4 (Gauss_Law_Drawings_8.gif)
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:
Figure 5
Figure 5 (Gauss_Law_Drawings_7.gif)
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:

Figure 6
Figure 6 (Gauss_Law_Drawings_10.gif)
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 ⃗ ⅆ V ∮ S F ⃗ ⋅ d a ⃗ = ∮ V ∇ ⃗ ⋅ F ⃗ ⅆ 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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