# Which are other anomalies like Divergence of 1/r^2?

As one might have learned in the basic science (ex. Electrodynamic theory), when we apply the divergence theorem to the vector function like 1/r^2 with it pointing in the radial direction (like electric filed of a point charge) the direct divergence gives value zero and the integration over a closed surface gives a non zero value. Which violates the divergence theorem. The explanation given is that the function blows at the origin(Delta function).

Can anyone please point out other anomalies like this (like in this case violation of the divergence theorem) in the physics or maths?

It is very important for me and will be really helpful.

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## 1 Answer

This isn't a violation of the divergence theorem, per se. In physics, we define: $\nabla \cdot \frac{\hat{r}}{r^2} = 4\pi \mathbf{\delta}^3({\mathbf{r}})$ since the definition of divergence is $\nabla \cdot \mathbf{F} = \displaystyle \lim_{V \rightarrow 0} \frac{\oint \mathbf{F}\cdot d\mathbf{a}}{V}$ [1]

So are you then asking about applications of the delta function in physics?

(I feel that this would be more appropriate to comment on your question rather than answer, but I don't have 50 reputation yet :( )

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Thanks for the answer. Yes, one can say the applications of delta function in physics. – Ruchit Rami Jun 5 '12 at 17:14
user, delta-functions are so standard and crucial in physics that they don't deserve to be called anomalies; they're business-as-usual. In quantum mechanics, delta-function appears in the inner product of eigenvectors in the presence of a continuous spectrum; in field theory, both classical and quantum, they appear on the right hand side of equations for Green's functions. Those examples are important because of the fact that any function may be written as a linear combination of many delta-functions at all possible points. Modern physics would be much less effective without delta-functions. – Luboš Motl Jun 5 '12 at 17:28
There are a whole bunch...the entire theory of Green's functions - basically reducing the solution of some nonhomogenous differential equations down to an summation over a collection of unit responses. We can model any sort of impulse localized to a particular time or place on an object as a delta function and see how the system responds to that impulse. Also of note is the Heaviside function, which is the integral of the delta function, defined to be 0 for t < 0 and 1 for t > 0. I'd suggest pretty much any decent text on mathematical physics as a place for examples since there are so many. – user758556 Jun 5 '12 at 17:30