The divergence theorem value for the function x2 + y2 + z2 at a distance of one unit from the origin is

To apply the divergence theorem to the function (f(x, y, z) = x^2 + y^2 + z^2) at a distance of one unit from the origin, we first need to understand what the divergence theorem is and how it’s used in this context.

The divergence theorem, also known as Gauss’s theorem, relates the flow (flux) of a vector field through a closed surface to the divergence of the vector field inside the volume enclosed by the surface. Mathematically, it can be expressed as:

[

intint_{Sigma} vec{F} cdot dvec{A} = intintint_{V} nabla cdot vec{F} dV

]

where (Sigma) is the closed surface, (V) is the volume enclosed by (Sigma), (vec{F}) is a vector field, (dvec{A}) is a vector element of area on (Sigma), and (dV) is an element of volume inside (V).

For the function (f(x, y, z) = x^2 + y^2 + z^2), we first convert this scalar function into a vector field. A common way to do this, given the context is not explicitly directing towards a specific vector field construction, is to consider a gradient field derived from (f), since the divergence theorem