Divergence Theorem Examples

These notes cover various examples of the Divergence Theorem, showcasing how to calculate flux through different three-dimensional shapes. Each example highlights unique properties and symmetries, making it easier to understand divergence in complex scenarios within Calculus IV.

1. Spherical shell

   • Represents a three-dimensional region between two concentric spheres.
   • Useful for calculating flux through a closed surface using the Divergence Theorem.
   • The divergence of a radial vector field can simplify calculations due to symmetry.

2. Solid cube

   • A simple geometric shape with equal sides, making it easy to apply the Divergence Theorem.
   • The volume integral can be computed easily due to constant limits of integration.
   • Provides a clear example of how to evaluate surface integrals on flat surfaces.

3. Cylindrical shell

   • Defined by a radius and height, allowing for cylindrical coordinates to simplify integration.
   • Useful for problems involving rotational symmetry and vector fields.
   • The divergence can be evaluated in cylindrical coordinates, making calculations more straightforward.

4. Paraboloid

   • A curved surface that can be oriented in various ways, affecting the divergence calculations.
   • Often used in problems involving gravitational fields or potential energy.
   • The divergence can be computed using cylindrical or Cartesian coordinates depending on orientation.

5. Torus

   • A doughnut-shaped surface that presents unique challenges in divergence calculations.
   • Requires careful consideration of the parameterization for surface integrals.
   • Useful for demonstrating the application of the Divergence Theorem in non-convex shapes.

6. Hemisphere

   • A half-sphere that simplifies calculations due to its symmetry.
   • Often used in problems involving flux through closed surfaces.
   • The divergence can be evaluated easily using spherical coordinates.

7. Cone

   • A three-dimensional shape with a circular base tapering to a point, useful for illustrating divergence in non-uniform fields.
   • The divergence can be computed using cylindrical coordinates, especially for vector fields aligned with the cone's axis.
   • Provides insight into how divergence behaves near singular points.

8. Ellipsoid

   • A stretched sphere that complicates divergence calculations due to its non-uniform shape.
   • Useful for demonstrating the application of the Divergence Theorem in more complex geometries.
   • Requires careful parameterization and integration limits when evaluating volume and surface integrals.

9. Tetrahedron

   • A polyhedron with four triangular faces, providing a simple yet effective example for divergence calculations.
   • The volume integral can be computed using straightforward limits in Cartesian coordinates.
   • Useful for illustrating the Divergence Theorem in a compact, easily visualizable shape.

10. Rectangular prism

    • A three-dimensional box shape that simplifies the application of the Divergence Theorem.
    • The constant limits of integration make volume and surface integrals straightforward to compute.
    • Serves as a foundational example for understanding divergence in Cartesian coordinates.