The divergence theorem states that the triple integral of the divergence of a vector field over a region is equal to the surface integral of a vector field over the boundary of , called .

For the divergence theorem the solid region doesn't need to be connected.

Examples

Example 1

Verify the divergence theorem with over the cube .

Split the cube into 6 different faces

Starting with the top face with the normal vector

By symmetry we can see that the same is true for all the sides.

Example 2

Use the divergence theorem to calculate the surface integral where and is the surface of the tetrahedron with vertices , , and with outward orientation.

Example 3

Compute where where is the upper unit hemisphere with upward pointing normal.

Consider as the circle disk with radius 1 defined around the origin in the -plane with downward orientation and the volume defined as .