Let be a fixed bounded region of space whose boundary surface is oriented outward

PHYSICS · Multi-part · Fermi University Physics · Mathematical Methods for Physics · Tensors and differential geometry · Tensors and differential geometry

Invariance, Ch Integrating forms: Stokes's theorem, holes and Maxwell's equations

Problem

Let DD be a fixed bounded region of space whose boundary surface ∂D\partial D is oriented outward, and let Q(t)=∫Dρ dVQ(t)=\int_D\rho\,\mathrm{d}V be the charge inside it. In spacetime with x0=ctx^0=ct the current three-form is ⋆J=cρ dx∧dy∧dz−Jx dx0∧dy∧dz−Jy dx0∧dz∧dx−Jz dx0∧dx∧dy,\begin{aligned}{\star}J ={}& c\rho\,\mathrm{d}x\wedge\mathrm{d}y\wedge\mathrm{d}z - J_x\,\mathrm{d}x^0\wedge\mathrm{d}y\wedge\mathrm{d}z\\ &- J_y\,\mathrm{d}x^0\wedge\mathrm{d}z\wedge\mathrm{d}x - J_z\,\mathrm{d}x^0\wedge\mathrm{d}x\wedge\mathrm{d}y,\end{aligned} and it satisfies d⋆J=0\mathrm{d}{\star}J=0. Take the four-dimensional region M=D×[t1,t2]M=D\times[t_1,t_2], oriented by dx0∧dx∧dy∧dz\mathrm{d}x^0\wedge\mathrm{d}x\wedge\mathrm{d}y\wedge\mathrm{d}z. ((a)) The boundary ∂M\partial M has three pieces: DD at time t2t_2, DD at time t1t_1, and the side ∂D×[t1,t2]\partial D\times[t_1,t_2]. Find the induced orientation of each piece. ((b)) Pull ⋆J{\star}J back to each piece and apply the generalised Stokes theorem to d⋆J\mathrm{d}{\star}J on MM to show that Q(t2)−Q(t1)=−∫t1t2(∮∂DJ⋅dS) dt.Q(t_2)-Q(t_1)=-\int_{t_1}^{t_2}\Big(\oint_{\partial D}\mathbf{J}\cdot\mathrm{d}\mathbf{S}\Big)\,\mathrm{d}t. ((c)) State the result in words.

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