Chapter 2 · Section 2.4
2.4 The Integral Theorems
When the gradient, the divergence or the curl is added up over a curve, a volume or a surface, everything inside cancels and only the boundary of the region survives.
- the fundamental theorem for gradients and why it makes line integrals of a gradient independent of the path,
- the divergence (Gauss's) theorem: outward flux through a closed surface volume integral of the divergence,
- Stokes' theorem: circulation around a closed curve flux of the curl through any surface spanning it, with the right-hand rule for orientations,
- why all three theorems are true (adding up small pieces whose inner contributions cancel), and
- how to verify a theorem in a concrete case, and the two identities and .
We now turn from derivatives to integrals. The gradient, the divergence and the curl were defined in Sections 2.2 and 2.5 as "change per unit length", "flux per unit volume" and "circulation per unit area" of something infinitesimally small. The integral theorems answer the natural follow-up question: what happens when we add up these local quantities over a region of finite size? The answer is always the same and very simple: everything inside cancels, and only the boundary of the region survives.
These theorems are the bridge between the two languages of electromagnetics. Laws such as Gauss's law can be stated either at a point (with a divergence) or for a whole region (with a flux), and the integral theorems translate one into the other. We meet three of them, in order of increasing dimension of the region: a curve, a volume and a surface.
The one-dimensional prototype is the fundamental theorem of calculus,
integrating a derivative over an interval gives the values of the function at the two ends of the interval. Keep this formula in mind, because each theorem of this section is a version of it.