While solving for the curl of the magnetic field($\vec \nabla\times \vec B = \mu_0 \vec J$), I got one formula which is written as $$\vec \nabla \cdot \frac{\hat r}{r^2} = 4\pi \delta^3 \vec r \tag{1}$$ and while solving for the verification of the Poisson's equation, I found the formula as follows $$\vec \nabla^2\frac{1}{\mid{\vec r - \vec r\:'}\mid} = -4 \pi \delta^3 (\vec r \:-\:\vec r\:')\tag{2}$$What are these equations called? How do we get these equations (1) and (2)? Can you please provide me the proof of these equations?
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Possible duplicates: https://physics.stackexchange.com/q/75557/2451 , https://physics.stackexchange.com/q/9255/2451 , https://math.stackexchange.com/q/368155/11127 and links therein. – Qmechanic Apr 28 '20 at 11:11
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https://physics.stackexchange.com/users/2451/qmechanic Sorry, I didn't really get my doubts cleared from the links you provided, I have searched many books for its proof, but failed to get any. I got to know that these equations are some formulas of the Dirac delta function, but I wanted the proof as Griffiths also didn't provide that proof. Please can you just provide me with the proof if you know? Thanks in advance. – Jack May 03 '20 at 19:07
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I added 1 more link. – Qmechanic May 03 '20 at 19:54
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Thank you very much. The new link helped me. – Jack May 04 '20 at 05:37