Question

For the vector field e = r¹⁰e-r - z³z, verify the divergence theorem for the cylindrical region enclosed by r = 2, z = 0, and z = 4?

Answer 1

To verify the divergence theorem, calculate the flux of the vector field through the cylindrical surface and compare it with the volume integral of the divergence of the vector field.

Step-by-step explanation:

To verify the divergence theorem for the given vector field e = r¹⁰e-r - z³z, we need to calculate the flux of the vector field through the cylindrical region enclosed by r = 2, z = 0, and z = 4 and compare it with the volume integral of the divergence of the vector field.

1. First, calculate the flux of the vector field through the cylindrical surface by integrating the dot product of the vector field and the outward unit normal vector over the surface.
2. Next, calculate the volume integral of the divergence of the vector field over the cylindrical region.
3. Finally, compare the two results. If they are equal, then the divergence theorem is verified.