Electromagnetic Fields and Waves: Notes, Solved Problems and Interactive Lab

2.4.5 Recap

Recap

Gradient theorem (2.4.1): $\int_{a}^{b} \nabla f \cdot d ℓ = f (b) - f (a)$ , independent of path, and $\oint \nabla f \cdot d ℓ = 0$ .
Divergence theorem (2.4.2): $\oint_{S} A \cdot d S = \int_{V} \nabla \cdot A d V$ , $d S$ outwards, $A$ smooth inside $V$ .
Stokes' theorem (2.4.3): $\oint_{C} A \cdot d ℓ = \int_{S} (\nabla \times A) \cdot d S$ , right-hand rule, any surface spanning $C$ .
Why: divide the region into small cells/patches. Shared faces/edges cancel in pairs, and only the boundary survives.
Consequences: $\nabla \times (\nabla f) = 0$ and $\nabla \cdot (\nabla \times A) = 0$ .
Singular fields ( $a_{r} / r^{2}$ , $a_{ϕ} / ρ$ ) need care: their flux or circulation can be non-zero although the divergence or curl vanishes everywhere except at the singular point or line.