Vector Calculus | Part IA, 2005

Let A(t,x)\mathbf{A}(t, \mathbf{x}) and B(t,x)\mathbf{B}(t, \mathbf{x}) be time-dependent, continuously differentiable vector fields on R3\mathbb{R}^{3} satisfying

At=×B and Bt=×A\frac{\partial \mathbf{A}}{\partial t}=\nabla \times \mathbf{B} \quad \text { and } \quad \frac{\partial \mathbf{B}}{\partial t}=-\nabla \times \mathbf{A}

Show that for any bounded region VV,

ddt[12V(A2+B2)dV]=S(A×B)dS\frac{d}{d t}\left[\frac{1}{2} \int_{V}\left(\mathbf{A}^{2}+\mathbf{B}^{2}\right) d V\right]=-\int_{S}(\mathbf{A} \times \mathbf{B}) \cdot d \mathbf{S}

where SS is the boundary of VV.

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