GATE 2014 EC (EC2) – Question 39
Let $\mathbf r=x\hat a_x+y\hat a_y+z\hat a_z$ and $r=|\mathbf r|$. Evaluate $\operatorname{div}(r^2\nabla\ln r)$.
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Correct answer: 2.9 to 3.1
Explanation
For r=|r⃗|>0, ∇lnr=r⃗/r². Multiplying by r² yields r⃗=x x̂+y ŷ+z ẑ. Its divergence is ∂x/∂x+∂y/∂y+∂z/∂z=**3**. The logarithm is undefined at r=0, so the calculation applies away from the origin.