1. In this question you will prove the neat reformulation of Maxwell's equations in terms of differential forms in R4 that I mentioned in class. Note that it does not matter whether you are in physics or not, as the calculations do not rely on any prior physics knowledge. We will do the calculation in two parts. In Part 1 we will show that the equation dF = 0 recovers two of Maxwell's equations. In Part 2, which is optional, we will show that the equation d(F) = J recovers the other two Maxwell's equations. Maxwell's equations, which govern electromagnetism, are vector calculus equations for the time-dependent electric and magnetic vector fields E and B:1 E =, B =0, E + B =0, t B E =J, t where J is a vector field representing the electric current density, and is the electric charge. Our goal is to reformulate these equations in terms of differential forms living in spacetime R4, with coordinates (t, x, y, z). We write E = (Ex, Ey, Ez), B = (Bx, By, Bz), and J = (Jx, Jy, Jz) for the time-dependent vector fields in R3. Note that for all of those, the component functions are general smooth functions in space R3 that also depend on time; for instance, Ex = Ex(t, x, y, z). We construct a two-form F in R4 that combines the electric field E and magnetic field B as follows: F =Bx dydz+By dzdx+Bz dxdyEx dtdxEy dtdyEz dtdz. We also construct a three-form J that combines the electric current density J and electric charge as follows: J = dx