Gauss's theorem, also known as Gauss's divergence theorem or Gauss's flux theorem, is a fundamental theorem in vector calculus. It relates the flux of a vector field through a closed surface to the divergence of the vector field within the volume enclosed by the surface. The theorem uses the operation of divergence to establish this relationship.

The divergence of a vector field is a scalar quantity that represents the amount of "outwardness" or "inwardness" of the vector field at a given point. It measures the rate at which the vector field is spreading out or converging at that point. Mathematically, the divergence of a vector field F is denoted as div(F).

Now, let's break down the Gauss theorem and understand how it uses the divergence operation:

1. Statement of Gauss's theorem:
The Gauss theorem states that the flux of a vector field F through a closed surface S is equal to the volume integral of the divergence of F within the region enclosed by S.

Mathematically, it can be expressed as:
∫∫∫(div(F)) dV = ∫∫(F · dA)

where ∫∫∫ represents the volume integral over the region enclosed by the surface, and ∫∫ represents the surface integral over the closed surface.

2. Flux and divergence:
The flux of a vector field through a surface measures the amount of flow of the vector field across the surface. It is given by the dot product of the vector field and the infinitesimal area vector dA.

The divergence of a vector field at a point measures the net flow of the vector field per unit volume around that point. It is given by the dot product of the gradient operator (∇) and the vector field.

3. Relationship between flux and divergence:
Gauss's theorem establishes a relationship between the flux of a vector field through a closed surface and the divergence of the vector field within the region enclosed by the surface.

The theorem states that the flux of a vector field through a closed surface is equal to the volume integral of the divergence of that vector field within the region enclosed by the surface.

In other words, the net flow of the vector field across the closed surface is determined by the divergence of the vector field within the enclosed region.

This relationship is expressed mathematically as ∫∫∫(div(F)) dV = ∫∫(F · dA), where div(F) represents the divergence of the vector field F.

Therefore, the correct answer is option 'C' - Divergence. The Gauss theorem uses the divergence operation to establish the relationship between the flux of a vector field through a closed surface and the divergence of the vector field within the enclosed region.