GAUSS'S THEOREM

1. Gauss's theorem

Gauss's theorem is also called Divergence theorem.
We will demonstrate it here.

Let's look at this figure. We consider that the box is small. So dx, dy and dz are small as well.

The net flux of the vector A at x is A(x) dy dz.
At x + dx, it's equal to A(x + dx) dy dz.

The variation of the vector A through the box from the surface (dy,dz) at x to the same surface at x+dx is:
A_x(x+dx) - A_x(x) = (∂A_x(x)/∂x) dx
Then the related net flux is:
(∂A_x(x)/∂x) dx dy dz = (∂A_x(x)/∂x) dV
Where dV = dx dy dz , the elementary volume.

With the same approach, we have :
Through the y axis:
(∂A_y(y)/∂y) dy dx dz = (∂A_y(y)/∂y) dV
Through the z axis:
(∂A_z(z)/∂z) dz dx dy = (∂A_z(z)/∂z) dV

The total flux out of the infinitesimal box is then:
dΦ_A = (∂A_x(x)/∂x + ∂A_y(x)/∂y + ∂A_z(z)/∂z))dV

The term ∂A_x(x)/∂x + ∂A_y(x)/∂y + ∂A_z(z)/∂z is the divergence of the vector A, that is ∇ . A.
So dΦ_A = ∇ . A dv
Hence :
Φ_A = ∫_{Volume: V} dΦ = ∫_V ∇ . A .dV
We know also that:
Φ_A = ∫_S A . dS
Where S is the surface that bounds the volume V Therefore, we get:

2. Particular case: The Electric field vector

In this case, we have:
∫∫_S E dS = Q/ε₀ (E-Gauss's law)
That is:
∫∫∫_V(∇.E)dV = ρ/ε₀∫∫∫_V dV
Thus:
∇.E = ρ/ε₀

3. Other result : The curl Theorem

If A = (A_x, A_y, A_z) and dv = dxdydz a volume element, then:
∫ ∇ x A dv = ∫[(∂A_y/∂z - ∂A_z/∂y)dxdydz,
(∂A_z/∂x - ∂A_x/∂z)dxdydz, (∂A_x/∂y - ∂A_y/∂x)dxdydz]
= ∫ [A_ydxdy - A_zdxdz, A_zdydz - A_xdxdy, A_xdxdz - A_ydydz]
= ∫ A x dS
with dS = (dydz, dxdz, dxdy) = ds. n
where n is the vector normal to the related surface. dydz, dxdz, dxdy are the faces perpendicular to their related normal vector.
Hence:
∫ ∇ x A dV = ∫ A x dS = ∫ A x n ds