Question

2. Using the vector identities listed in the book and the Divergence theorem with appropriate substitution, prove the following relations (Refer to section 1.4.3 in the textbook) \text { (a) } \int_{V} d^{3} r\left[\phi \nabla^{2} \psi+\nabla \phi \cdot \nabla \psi\right]=\int_{S} d \mathbf{d} \cdot \phi \nabla \psi \int_{V} d^{3} r\left[\phi \nabla^{2} \psi-\psi \nabla^{2} \phi\right]=\int_{S} d \mathbf{S} \cdot[\phi \nabla \psi-\psi \nabla \phi] \text { (c) } \int_{V} d^{3}+[\nabla \times \mathbf{P} \cdot \nabla \times \mathbf{Q}-\mathbf{P} \cdot \nabla \times \nabla \times \mathbf{\nabla}]=\int_{g} \mathrm{dS} \cdot(\mathbf{P} \times \nabla \times \mathbf{Q}) \int_{V} d^{3} r[Q \cdot \nabla \times \nabla \times \mathbf{P}-\mathbf{P} \cdot \nabla \times \nabla \times \mathbf{Q}]=\int_{S} \mathrm{dS} \cdot[\mathrm{P} \times \nabla \times \mathbf{Q}-\mathbf{Q} \times \nabla \times \mathbf{P}]

Fig: 1

Fig: 2

Fig: 3

Fig: 4

Fig: 5