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Divergence Theorem

Source lecture(s): PC604 Lec1-2

Intuition

Sources inside a volume push flux out through its surface.

Formal Definition

\[\oint_S \vec{A}\cdot d\vec{S}=\int_V (\nabla\cdot\vec{A})\,dV\]

Mathematical Formulation

\[\oint_S \vec{A}\cdot d\vec{S}=\int_V (\nabla\cdot\vec{A})\,dV\]

Derivation

Apply the fundamental theorem in each Cartesian direction and sum.

Worked Example

For \(\vec{A}=x\hat{i}\) through the unit cube, flux is \(1\), matching \(\int_V 1\,dV=1\).

Common Mistakes

  • Forgetting the outward normal on \(d\vec{S}\).
  • Applying to non-closed surfaces.

Quiz

Q1. Can the divergence theorem be applied to a hollow sphere?

Answer

Yes—include both inner and outer surfaces.