Calculus III - Divergence Theorem

Introduction to Divergence Theorem

• Purpose: Relates surface integrals to triple integrals.
• Key concept: The Divergence Theorem connects the flow of a vector field across a surface to the behavior of the vector field inside the volume bounded by the surface.

The Divergence Theorem

• Given:
  • E: A simple solid region.
  • S: The boundary surface of E with positive orientation.
  • F: A vector field whose components have continuous first order partial derivatives.
• Theorem Statement: [ \int_{S} \mathbf{F} \cdot d\mathbf{S} = \int_{E} \text{div} \mathbf{F} , dV ]
  • The left side represents the surface integral of vector field F over surface S.
  • The right side represents the triple integral of the divergence of F over the volume E.

Example Problem

• Objective: Use the Divergence Theorem to evaluate the integral.
• Vector Field (F):
  • [ F = x y \mathbf{i} + \frac{1}{2} y^2 \mathbf{j} + z \mathbf{k} ]
• Description of Surface:
  • Three surfaces make up the boundary:
    • Top Surface: [ z = 4 - \frac{3}{2}(x^2 + y^2) ]
    • Side Surface: [ x^2 + y^2 = 1 ]
    • Bottom Surface: [ z = 0 ]
• Solution Approach:
  • Apply the Divergence Theorem to calculate the integral over the defined surfaces.

Conclusion

• The Divergence Theorem is a powerful tool in vector calculus that simplifies the computation of surface integrals by converting them into volume integrals.
• Particularly useful for fields with symmetry or where direct computation of surface integrals is complex.