Divergence Theorem:
The divergence theorem relates a volume integral of a vector field to a surface integral of the vector field over the bounding surface of the volume. It states that the flux of a vector field across a closed surface is equal to the volume integral of the divergence of the vector field over the enclosed volume.

Given Vector Field:
The given vector field is F = (x^2, y^2, z^2).

Divergence of the Vector Field:
To calculate the divergence of the vector field, we need to find the partial derivatives of each component with respect to their corresponding variables and sum them up.

∇ · F = ∂(x^2)/∂x + ∂(y^2)/∂y + ∂(z^2)/∂z
= 2x + 2y + 2z

Application of Divergence Theorem:
Using the divergence theorem, we can evaluate the flux of the vector field across a closed surface by calculating the volume integral of the divergence over the enclosed volume.

In this case, the vector field is evaluated at a distance of one unit from the origin, which means we are considering a sphere centered at the origin with a radius of one unit.

Volume Integral:
To calculate the volume integral, we need to find the triple integral of the divergence of the vector field over the volume of the sphere.

∫∫∫ (2x + 2y + 2z) dV

Spherical Coordinates:
To evaluate the triple integral, it is convenient to switch to spherical coordinates since we are dealing with a sphere.

∫∫∫ (2ρsinφcosθ + 2ρsinφsinθ + 2ρcosφ) ρ^2sinφ dρ dθ dφ

Limits of Integration:
The limits of integration for ρ, θ, and φ are as follows:

ρ: 0 to 1 (since we are considering the sphere of radius one unit)
θ: 0 to 2π (full revolution around the z-axis)
φ: 0 to π (from the positive z-axis to the negative z-axis)

Integration:
Evaluating the triple integral, we get:

∫∫∫ (2ρsinφcosθ + 2ρsinφsinθ + 2ρcosφ) ρ^2sinφ dρ dθ dφ

= ∫[0 to 2π] ∫[0 to π] ∫[0 to 1] (2ρ^3sin^2φcosθ + 2ρ^3sin^2φsinθ + 2ρ^3sinφcosφ) dρ dθ dφ

= 2/4 * 2π * 2 * π * 1^4 * 1/4 + 0 + 0

= 2π/2

= π

Therefore, the divergence theorem value for the given vector field at a distance of one unit from the origin is π, which is approximately 3.