• Danko Antolovic


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 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:
$$\overline{\varphi } = \frac{1} {{a}^{3}}\quad { \int \nolimits }_{a}\varphi (\mathbf{x})\mathrm{d}{x}_{1}\mathrm{d}{x}_{2}\mathrm{d}{x}_{3}$$
We expand the field in a Taylor series around the origin:
$$\varphi (\mathbf{x}) = \varphi (0) + \sum \limits_{i}{\left ( \frac{\partial \varphi } {\partial {x}_{i}}\right )}_{0}{x}_{i} + \frac{1} {2} \sum \limits_{i,j}{\left ( \frac{{\partial }^{2}\varphi } {\partial {x}_{i}\partial {x}_{j}}\right )}_{0}{x}_{i}{x}_{j} + \cdots $$
and substitute the series into the integral in (10.1). Integration over any odd power of x i yields zero; therefore, all linear terms vanish, as well as all quadratic terms for which ij. 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:
$$\bar{\varphi } - \varphi (0) = \frac{{a}^{2}} {24}{\left ({\nabla }^{2}\varphi \right )}_{ 0}$$
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.


Transmission Line Medium Access Control Layer Reciprocity Theorem Power Flux Control Frame 
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© Springer Science+Business Media, LLC 2010

Authors and Affiliations

  1. 1.University Information Technology ServicesIndiana UniversityBloomingtonUSA

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