Divergence Theorem for a Surface Consisting of a Sphere

The divergence theorem, also known as Gauss's theorem, relates the flux of a vector field through a closed surface to the divergence of the vector field within the volume enclosed by the surface. It is a fundamental theorem in vector calculus.

In the context of the given question, the surface under consideration is a sphere. To apply the divergence theorem, we need to determine the appropriate coordinate system in which the theorem can be computed.

1. Cartesian Coordinate System:
The Cartesian coordinate system is defined by three perpendicular axes (x, y, and z) intersecting at the origin. It is commonly used for three-dimensional geometry and vector calculus problems. In this coordinate system, the divergence theorem can be applied to calculate the flux through a closed surface enclosing a volume.

2. Cylindrical Coordinate System:
The cylindrical coordinate system is defined by a radial distance (ρ), azimuthal angle (φ), and height (z). It is particularly useful for problems with cylindrical symmetry. However, the divergence theorem is not typically applied directly in this coordinate system for a surface consisting of a sphere.

3. Spherical Coordinate System:
The spherical coordinate system is defined by a radial distance (r), polar angle (θ), and azimuthal angle (φ). It is commonly used for problems with spherical symmetry. When dealing with a surface consisting of a sphere, the spherical coordinate system is the most appropriate choice. The divergence theorem can be expressed in terms of the spherical coordinate system to compute the flux through the spherical surface enclosing the volume.

4. Depends on the Function:
In some cases, the choice of coordinate system may depend on the specific vector field or function being analyzed. While the spherical coordinate system is typically suitable for a surface consisting of a sphere, certain vector fields or functions may require a different coordinate system to accurately apply the divergence theorem.

In summary, the divergence theorem for a surface consisting of a sphere is computed in the spherical coordinate system. However, the choice of coordinate system may vary depending on the specific vector field or function involved in the problem.