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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