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

Looking back, the three theorems of this section are one idea expressed in three dimensions of "region": the integral of a derivative over a region equals the integral of the function itself over the boundary of that region. The fundamental theorem for gradients relates a curve to its two endpoints, Stokes' theorem relates a surface to its closed rim, and the divergence theorem relates a volume to its closed surface (Figure 2.4.11 and Table 2.4.1). In each case the derivative involved is precisely the one whose definition is "change per unit length", "circulation per unit area" or "flux per unit volume", and each proof consists of adding up infinitesimal pieces whose internal contributions cancel.

Gradient theorem: a curve \to its two endpoints, \int_{\vec a}^{\vec b}(\nabla f)\cdot d\vec\ell=f(\vec b)-f(\vec a) — Gradient theorem: a curve $\to$ its two endpoints, $\int_{a}^{b} (\nabla f) \cdot d ℓ = f (b) - f (a)$

Divergence theorem: a volume \to its closed surface, \oint_S\vec A\cdot d\vec S=\int_V(\nabla\cdot\vec A)\,dV — Divergence theorem: a volume $\to$ its closed surface, $\oint_{S} A \cdot d S = \int_{V} (\nabla \cdot A) d V$

Stokes' theorem: a surface \to its closed rim, \oint_C\vec A\cdot d\vec\ell=\int_S(\nabla\times\vec A)\cdot d\vec S. The three integral theorems side by side. In each, the integral of a derivative over a region (curve, volume, surface) equals an integral of the function itself over the boundary of that region (two endpoints, closed surface, closed curve). — **Figure 2.4.11.** Stokes' theorem: a surface $\to$ its closed rim, $\oint_{C} A \cdot d ℓ = \int_{S} (\nabla \times A) \cdot d S$ . The three integral theorems side by side. In each, the integral of a derivative over a region (curve, volume, surface) equals an integral of the function itself over the boundary of that region (two endpoints, closed surface, closed curve).

Table 2.4.1. The three integral theorems compared.

	gradient theorem	divergence theorem	Stokes' theorem
region	curve $C$	volume $V$	open surface $S$
boundary	two endpoints $a$ , $b$	closed surface $S$	closed curve $C$ (rim)
derivative	$\nabla f$	$\nabla \cdot A$	$\nabla \times A$
meaning of the derivative	change per unit length	flux per unit volume	circulation per unit area
orientation	from $a$ to $b$	$d S$ outwards	right-hand rule between $C$ and $d S$
consequence	$\oint \nabla f \cdot d ℓ = 0$ , path independence	divergence-free $\Rightarrow$ zero net flux	curl-free $\Rightarrow$ zero circulation, same flux for every surface with rim $C$
in electrostatics	potential difference from $E$ (Section 3.3)	Gauss's law, integral $\leftrightarrow$ differential (Section 3.2)	$\nabla \times E = 0$ , conservative field (Section 3.3)

These theorems will be used constantly. In Chapter 3 (Sections 3.2 to 3.4) they translate Gauss's law between its integral and differential forms, establish that the electrostatic field is conservative, and let us compute potentials as line integrals of the field. Later in the course they do the same for all four of Maxwell's equations, which are statements about two divergences and two curls.

2.4.4 The three integral theorems