Form-Invariance of Maxwell Equations in Integral Form Cristian E. Gutiérrez To Richard Wheeden on the occasion of his retirement Abstract We find transformation formulas for weak solutions to Maxwell’s equa- tions in integral form by general changes of coordinates obtaining that the equations are also “form invariant” as in the standard case. Solutions are defined using test functions. 1 Introduction In this note we consider weak solutions to Maxwell equations and show their invariance under changes of coordinates. These changes are assumed to be locally Lipschitz functions and therefore they might be not differentiable in a set of Lebesgue measure zero. In this formulation, the fields E and H and the permittivity .x/ and the permeability .x/ might be discontinuous and only need to satisfy Lebesgue integrability conditions. From the invariance, we recover the remarkable fact, that standard Maxwell’s equations preserve their form under smooth coordinate transformations, see [9] and [7, Chap. 5]. 1 1 I would like to thank Professor Ulf Leonhardt for pointing out a connection between his paper [6, Sect. 3.2] with our results. I also like to thank the referee for pointing out a connection between our results and Maxwell’s vacuum equations in general relativity, and for providing reference [1, Chap. 10, Sect. 11]. C.E. Gutiérrez () Department of Mathematics, Temple University, Philadelphia, PA 19122, USA e-mail: gutierre@temple.edu; c.e.a.gutierrez@gmail.com © Springer International Publishing AG 2017 S. Chanillo et al. (eds.), Harmonic Analysis, Partial Differential Equations and Applications, Applied and Numerical Harmonic Analysis, DOI 10.1007/978-3-319-52742-0_5 69