Module 1 : A Crash Course in Vectors
Lecture 6 : Laplacian
  Example :
  Evaluate $\int_0^5\cos x \delta(x-\pi)dx$.
  Solution :
  The range of integration includes the point $x= \pi$ at which the argument of the delta function vanishes. Thus, the value of the integral is $\cos\pi =-1$.
  Exercise :
  Evaluate $\int \vec r\cdot (\vec a-\vec r)\delta(\vec r-\vec b)d^r$, where $\vec a = (1,2,3), \vec b = (3,2,1)$ and the integration is over a sphere of radius 1.5 centered at (2,2,2)(Ans. -4)  A physical example is the volume density of charge in a region which contains a point charge $q$. The charge density is zero everywhere except at the point where the charge is located. However, the volume integral of the density in any region which includes this point is equal to $q$ itself. Thus if $q$ is located at the point $\vec r = \vec a$, we can write
 
\begin{displaymath}\rho(\vec r) = q\delta(\vec r -\vec a)\end{displaymath}
   
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