# Appendices

Chapter

First Online:

## Abstract

As we said in Sect. 1.1, Laplacian relates the value of a field at a point in space to the field’s average value around that point. To see this, let us imagine a small cube with side We expand the field in a Taylor series around the origin: and substitute the series into the integral in (10.1). Integration over any odd power of Difference between the average and the central value of φ decreases with the size of the cube, as it must, but for a given cube size, it is proportional to the value of the field’s Laplacian at the center.

*a*, centered around the given point; we will also make this point the coordinate origin, for simplicity. The average value of a scalar field φ over the cube is:$$\overline{\varphi } = \frac{1} {{a}^{3}}\quad { \int \nolimits }_{a}\varphi (\mathbf{x})\mathrm{d}{x}_{1}\mathrm{d}{x}_{2}\mathrm{d}{x}_{3}$$

(10.1)

$$\varphi (\mathbf{x}) = \varphi (0) + \sum \limits_{i}{\left ( \frac{\partial \varphi } {\partial {x}_{i}}\right )}_{0}{x}_{i} + \frac{1} {2} \sum \limits_{i,j}{\left ( \frac{{\partial }^{2}\varphi } {\partial {x}_{i}\partial {x}_{j}}\right )}_{0}{x}_{i}{x}_{j} + \cdots $$

(10.2)

*x*_{ i }yields zero; therefore, all linear terms vanish, as well as all quadratic terms for which*i*≠*j*. Integration over zero-th power of*x*_{ i }yields*a*, the dimension of the cube, and integration over*x*_{ i }^{2}yields factors proportional to*a*^{3}. The end result is:$$\bar{\varphi } - \varphi (0) = \frac{{a}^{2}} {24}{\left ({\nabla }^{2}\varphi \right )}_{ 0}$$

(10.3)

## Keywords

Transmission Line Medium Access Control Layer Reciprocity Theorem Power Flux Control Frame
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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