2.6: Integral Theorems

Several integral theorems prove useful, not only in the description of electromagnetic fields but also in dealing with continuum mechanics and electromechanics. These theorems will be stated here without proof.

If it is recognized that the gradient operator is defined such that its line integral between two endpoints (a) and (b) is simply the scalar function evaluated at the endpoints, then\(^{1}\)

\[ \int_{\overrightarrow{a}}^{\overrightarrow{b}} \nabla \psi \cdot \overrightarrow{d}l = \psi (\overrightarrow{b}) - \psi (\overrightarrow{a})\ \label{1} \]

Two more familiar theorems\(^{1}\) are useful in dealing with vector functions. For a closed surface \(S\), enclosing the volume \(V\), Gauss' theorem states that

\[ \int_{V} \nabla \cdot \overrightarrow{A} dV = \oint_{S} \overrightarrow{A} \cdot \overrightarrow{n} da \label{2} \]

while Stokes's theorem pertains to an open surface \(S\) with the contour \(C\) as its periphery:

\[ \int_{S} \nabla \times \overrightarrow{A} \cdot \overrightarrow{n} da = \oint_{C} \overrightarrow{A} \cdot \overrightarrow{d}l \label{3} \]

In stating these theorems, the normal vector is defined as being outward from the enclosed voluge for Gauss' theorem, and the contour is taken as positive in a direction such that \(\overrightarrow{d}l\) is related to \(\overrightarrow{n}\) by the right-hand rule. Contours, surfaces, and volumes are sketched in Figure 2.6.1.

A possibly less familiar theorem is the generalized Leibnitz rule.\(^{2}\) In those cases where the surface is itself a function of time, it tells how to take the derivative with respect to time of the integral over an open surface of a vector function:

Figure 2.6.1. Arbitrary contours, volumes and surfaces: (a) open contour \(C\); (b) closed surface \(S\), enclosing volume \(V\); (c) open surface \(S\) with boundary contour \(C\).

\[ \frac{d}{dt} \int_{S} \overrightarrow{A} \cdot \overrightarrow{n} da = \int_{S} \big[ \frac{\partial{\overrightarrow{A}}}{\partial{t}} + (\nabla \cdot \overrightarrow{A}) \overrightarrow{v_s}\big] \cdot \overrightarrow{n} da + \oint_{C} (\overrightarrow{A} \times \overrightarrow{v}) \cdot \overrightarrow{d}l \label{4} \]

Again, \(C\) is the contour which is the periphery of the open surface \(S\). The velocity vs is the velocity \(\overrightarrow{v_s}\) of the surface and the contour. Unless given a physical significance, its meaning is purely geometrical.

A limiting form of the generalized Leibnitz rule will be handy in dealing with closed surfaces Let the contour \(C\) of Equation \ref{4} shrink to zero, so that the surface \(S\) becomes a closed one. This process can be readily visualized in terms of the surface and contour sketch in Figure 2.6.1c if the contour \(C\) is pictured as the draw-string on a bag. Then, if \(\xi \equiv \nabla.\overrightarrow{A}\), and use is made of Gauss' theorem (Equation \ref{2}), Equation \ref{4} becomes a statement of how to take the time derivative of a volume integral when the volume is a function of time:

\[ \frac{d}{dt} \int_{V} \xi dV = \int_{V} \frac{\partial{\xi}}{\partial{t}} dV + \oint_{S} \xi \overrightarrow{v_s} \cdot \overrightarrow{n} da \label{5} \]

Again, \(\overrightarrow{v_s}\) is the velocity of the surface enclosing the volume \(V\).

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1. Markus Zahn, Electromagnetic Field Theory, a problem solving approach, John Wiley & Sons, New York, 1979, pp. 18-36.

2. H. H. Woodson and J. R. Melcher, Electromechanical Dynamics, Vol. 1. John Wiley & Sons, New York, 1968, pp. B32-B36.(See Prob. 2.6.2 for the derivation of this theorem.)