By Thomas A. Garrity

This article is an creation to a couple of the mathematical wonders of Maxwell's equations. those equations ended in the prediction of radio waves, the conclusion that gentle is a kind of electromagnetic wave, and the invention of the specific concept of relativity. in truth, just about all present descriptions of the basic legislation of the universe will be seen as deep generalizations of Maxwell's equations. much more fabulous is that those equations and their generalizations have ended in one of the most very important mathematical discoveries of the prior thirty years. it sounds as if the maths in the back of Maxwell's equations is unending. The objective of this publication is to provide an explanation for to mathematicians the underlying physics in the back of electrical energy and magnetism and to teach their connections to arithmetic. beginning with Maxwell's equations, the reader is ended in such issues because the precise idea of relativity, differential kinds, quantum mechanics, manifolds, tangent bundles, connections, and curvature

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Choose any coordinate system t , x , y , and z in which the particle is moving at some constant v = (vx , v y , vz ) whose speed we also denote by v. The relativistic displacement vector is (c t , x , y , z ). In the different coordinate system that gives us the proper time (in which the particle is not moving at all), if t measures the change in proper time, we know from three sections ago that t = γ t. Then for the momentum we have p = ( pt , p x , p y , p z ) =m· c t , t x , t y , t z t =m· c t · t t , t x · t t , t = γ mc, γ m x ,γ m t y · t t , t z · t t t y z ,γ m t t = γ mc, γ mvx , γ mv y , γ mvz .

For this last problem, I simply chose, almost at random, φ(x, y, z, t) = x z and A = ( − yt + x 2 , x + zt 2 , −y + z 2 t). 16 2 Maxwell’s Equations The punchline of Chapter 7 is that the converse holds, meaning that if the function ρ and the vector fields E, B, and j satisfy Maxwell’s equations, then there must be a function φ(x, y, z, t) and a vector field A such that E = −∇(φ) − ∂∂tA and B = ∇ × A. The φ(x, y, z, t) and A are called the potentials. 3 Electromagnetic Waves Summary: When the current j and the density ρ are zero, both the electric field and the magnetic field satisfy the wave equation, meaning that both fields can be viewed as waves.

The question for this section is how to relate the vector v1 with the vector v2 . Since we have    t1 t2 x 1 (t1 ) x 2 (t2 )      y2 (t2 ) = A  y1 (t1 ) , z 2 (t2 ) z 1 (t1 )  we can compute, for example, that dx 2 d(x 2 as a function of (x 1 , y1 , z 1 , t1 )) dt1 = , dt2 dt1 dt2 allowing us to relate dx 2 /dt2 to dx 1 /dt1 , dy1 /dt1 and dz 1 /dt1 . In a similar way, we can find dy2 /dt2 and dz 2 /dt2 in terms of the dx 1 /dt1 , dy1 /dt1 and dz 1 /dt1 . To make this more concrete, let us consider one possible A, namely, the coordinate change x 2 = γ x 1 − γ vt1 y2 = y1 z2 = z1 t2 = −γ v x 1 + γ t1 , c2 44 4 Special Relativity where γ = 1/ 1 − ( vc )2 .

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