Question

Consider a vector function \(\vec{u}(\vec{r})\) and two scalar functions \(\psi(\vec{r})\) and \(\phi(\vec{r})\). The unit vector \(\hat{n}\) is normal to the elementary surface \(dS\), \(dV\) is an infinitesimal volume, \(d\vec{l}\) is an infinitesimal line element, and \(\partial / \partial n\) denotes the partial derivative along \(\hat{n}\). Which of the following identities is/are correct?

• A. \(\displaystyle \int_V \psi \, \nabla \cdot \vec{u} \, dV = \oint_S \psi \, \vec{u} \cdot \hat{n} \, dS\), where surface \(S\) bounds the volume \(V\).
• B. \(\displaystyle \int_V [\psi \nabla^2 \phi - \phi \nabla^2 \psi] dV = \oint_S \left[\psi \frac{\partial \phi}{\partial n} - \phi \frac{\partial \psi}{\partial n}\right] dS\), where \(S\) bounds \(V\).
• C. \(\displaystyle \int_V [\psi \nabla^2 \phi - \phi \nabla^2 \psi] dV = \oint_S \left[\psi \frac{\partial \phi}{\partial n} + \phi \frac{\partial \psi}{\partial n}\right] dS\), where \(S\) bounds \(V\).
• D. \(\displaystyle \oint_S \phi \, \vec{u} \cdot d\vec{l} = \iint_S (\nabla \times \vec{u}) \cdot \hat{n} \, dS\), where \(C\) is the boundary of \(S\).

Hint:

Green's identities are derived from the divergence theorem and are essential in vector calculus and electrostatics for relating surface and volume integrals.

Answer 1

The Correct Option is A, B, D

Step 1: Recall Green's second identity.
Green's second identity relates two scalar functions \(\psi\) and \(\phi\) over a volume \(V\) and its bounding surface \(S\): \[ \int_V (\psi \nabla^2 \phi - \phi \nabla^2 \psi) dV = \oint_S \left(\psi \frac{\partial \phi}{\partial n} - \phi \frac{\partial \psi}{\partial n}\right) dS \] This is a standard result derived using the divergence theorem.

Step 2: Check each option.
- (A) represents the divergence theorem, but lacks \(\nabla \psi\) terms, hence incomplete in this context. - (B) matches exactly with Green's second identity — correct. - (C) has a sign error — the plus sign makes it incorrect. - (D) confuses Stokes' theorem: it should relate a line integral of \(\vec{u}\) around \(C\) to \((\nabla \times \vec{u})\) over \(S\), but the expression given involves \(\phi\), hence incorrect.

Step 3: Conclusion.
Thus, only option (B) is correct.