Verify Formula (1) in the Divergence Theorem by evaluating...

Verify Formula (1) in the Divergence Theorem by evaluating the surface integral and the triple integral. $$ \begin{array}{l}{\mathbf{F}(x, y, z)=x y \mathbf{i}+y z \mathbf{j}+x z \mathbf{k} ; \sigma \text { is the surface of the cube }} \\ {\text { bounded by the planes } x=0, x=2, y=0, y=2, z=0} \\ {z=2}\end{array} $$

Verify Formula (1) in the Divergence Theorem by evaluating the surface integral and the triple integral.
$$
\begin{array}{l}{\mathbf{F}(x, y, z)=x y \mathbf{i}+y z \mathbf{j}+x z \mathbf{k} ; \sigma \text { is the surface of the cube }} \\ {\text { bounded by the planes } x=0, x=2, y=0, y=2, z=0} \\ {z=2}\end{array}
$$

Key Concepts

Triple Integral over a Volume

A triple integral is used to integrate a function over a three-dimensional region, summing contributions from infinitesimal volume elements. When applied to the divergence of a vector field, the triple integral represents the net 'source strength' within the volume, which according to the divergence theorem, equals the net flux across the boundary surface.

Surface Integral (Flux Integral)

A surface integral in the context of vector fields calculates the total flux across a surface by integrating the dot product of the vector field and the surface's unit normal vector over the entire surface. This measure is used to quantify how much of the field 'flows' out of or into the surface, and it forms one side of the equality established by the divergence theorem.

Divergence Theorem

The divergence theorem, a fundamental result in vector calculus, relates the flux of a vector field passing through a closed surface to the triple integral of the divergence of the field over the region bounded by that surface. It essentially converts a surface integral into a volume integral, thereby providing a powerful method for evaluating flux when the divergence is easier to integrate than the original field over the boundary.

Divergence of a Vector Field

The divergence of a vector field is a scalar measure of the net rate of 'expansion' or 'contraction' at a point in the field. It is calculated by taking the sum of the partial derivatives of each component of the vector field with respect to its corresponding coordinate. This concept is central to understanding how the flux through a closed surface is related to the behavior of the field inside the volume.

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