# Appendices

Chapter

## 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 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)
We expand the field in a Taylor series around the origin:
$$\varphi (\mathbf{x}) = \varphi (0) + \sum \limits_{i}{\left ( \frac{\partial \varphi } {\partial {x}_{i}}\right )}_{0}{x}_{i} + \frac{1} {2} \sum \limits_{i,j}{\left ( \frac{{\partial }^{2}\varphi } {\partial {x}_{i}\partial {x}_{j}}\right )}_{0}{x}_{i}{x}_{j} + \cdots$$
(10.2)
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:
$$\bar{\varphi } - \varphi (0) = \frac{{a}^{2}} {24}{\left ({\nabla }^{2}\varphi \right )}_{ 0}$$
(10.3)
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.

### Keywords

Microwave Sine Refraction Azimuth Coupler

### References

1. Abramowitz, M., Stegun, E.A. Handbook of Mathematical Functions, Dover, New York (1972)
2. Altmann, S.L. Rotations, Quaternions and Double Groups, Oxford University Press, Oxford (1986)
3. Elliott, R.S. Antenna Theory and Design, revised edition, Wiley, New York (2003)Google Scholar
4. Gast, M.S. 802.11 Wireless Networks, 2nd edition, O’Reilly, CA (2005)Google Scholar
5. Hobson, E.W. Spherical and Ellipsoidal Harmonics, 2nd reprint, Chelsea, New York (1965)Google Scholar
6. Horowitz, P., Hill, W. The Art of Electronics, 2nd edition, Cambridge University Press, Cambridge (1989)Google Scholar
7. Jackson, J.D. Classical Electrodynamics, 3rd edition, Wiley, New York (1998)Google Scholar
8. Messiah, A. Quantum Mechanics, Dover, New York (1999)Google Scholar
9. Pearson, B. Complementary Code Keying Made Simple, application note AN9850.1, Intersil Corporation, FL (2000)Google Scholar
10. Pozar, D.M. Microwave Engineering, Wiley, New York (1998)Google Scholar
11. Razavi, B. RF Microelectronics, Prentice Hall, NJ (1998)Google Scholar
12. Sklar, B. Digital Communications, 2nd edition, Prentice Hall, NJ (2001)Google Scholar
13. Talman, J.D. Special Functions: A Group Theoretic Approach, Benjamin, New York (1968)