Question

Consider a volume integral
$I=\int_V\nabla^2(\frac{1}{r})dv$
over a volume $V$, where $r = \sqrt{x^2 + y^2 + z^2}$. Which of the following statements is/are correct?

• A. The integrand vanishes for $r \neq 0$.
• B. The integrand vanishes for $r \neq 0$.
• C. $I = 0$, if $r = 0$ is not inside the volume $V$.
• D. The integrand diverges as $r \rightarrow \infty$.

Answer 1

Answer: (A), (B), (C)

The integrand vanishes for $r \neq 0$.

Answer 2

The correct options are (A) :$I = -4\pi$, if $r = 0$ is inside the volume $V$.(B):The integrand vanishes for $r \neq 0$.(C):$I = 0$, if $r = 0$ is not inside the volume $V$.