In this question you will prove the neat reformulation of Maxwell's equations in terms of differential forms in 4 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 ??=0 recovers two of Maxwell's equations. In Part 2, which is optional, we will show that the equation ?(?)=? 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 ? and ? : (I am using units in which the speed of light is ?=1 , which is standard in modern physics, and I have rescaled the electric charge and the electric current density to absorb a factor of 4? .) ??=??=?? ??=????=?,0,0,?, where ? 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 4 , with coordinates (?,?,?,?) . We write ?=(??,??,??) , ?=(??,??,??) , and ?=(??,??,??) for the time-dependent