Fundamental Solution of Poissons and Laplaces Equations

Now we return to Poisson s and Laplace s equations, which describe the behavior of the potential inside and outside masses, respectively. Earlier we have already derived an expression for the potential  

Taking the first and then the second derivatives, we find 3. = + 3( /. - XqfL  

By definition, the Laplacian of U represents the divergence of the attraction field, and, correspondingly, its value characterizes the density of masses at same point. Now the following question arises. What does the Laplacian tells us about the behavior of the potential To answer this question we first consider the simplest case, when U depends on one argument, x, Fig. 1.7a. Then, we can represent the derivatives as  

the first derivative defines the rate of change of the function U x), while the second derivative shows how its average value differs from the value of the function at the same point. For instance, if U x) Q, we have U(x) [/ (v) and there is a maximum. Next, suppose that within some interval of v the average value of the function coincides at each point with the value of this function  

The last equation represents the simplest class of functions in the one-dimensional case, namely, the linear functions, for which the condition (1.61) is met. Correspondingly, we can say that the second derivative is a measure of how the behavior 

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