The electric and magnetic fields as fixed-time spatial vector fields

Let the laboratory region $U\cong I\times \mathrm{\Omega }$, its spatial Euclidean structure and its units be those of Laboratory geometry: a restricted inertial chart, its spatial slices and fixed-time fields and Framework-local SI units, the vacuum constants and the exact/measured status of their values, and let $k\ge 1$. An electric field of class $C^k$ on the region is a map $E:\mathrm{\Omega }\times I\to {\mathbb{R}}^3$ that is of class $C^k$ in the sense of $C^k$ Euclidean maps and diffeomorphisms; a magnetic field of class $C^k$ is a map $B:\mathrm{\Omega }\times I\to {\mathbb{R}}^3$ of the same class. At fixed $t\in I$ the maps $E(\cdot ,t),B(\cdot ,t):\mathrm{\Omega }\to {\mathbb{R}}^3$ are real spatial vector fields on the region $\mathrm{\Omega }$, read in the $\delta$-orthonormal basis $(e_1,e_2,e_3)$ of the frame. The framework uses: $C^1$ fields in the classical Maxwell branch, $C^2$ fields where second or mixed time-space derivatives of the fields appear, and $C^{\mathrm{\infty }}$ fields in the smooth existence branches; every statement carries its own regularity and no regularity beyond it is asserted.

Conventions:

(a) The polar-vector and axial-vector conventions mean the following component rules under a spatial orthogonal coordinate change $y^{\mathrm{\prime }}=Qy$, where $Q$ is a real $3\times 3$ matrix with $Q^{\mathsf{T}}Q=I_3$. Transpose, matrix product and identity are those of The transpose $A^{\mathsf{T}}$ of a matrix and Rectangular matrix multiplication and the identity matrix $I_n$, including zero-sized shapes, and $\det Q$ is the determinant of For $n\ge 1$, the determinant over a commutative ring by the Leibniz formula, and $\mathrm{\mid }\det A\mathrm{\mid }$ for a real matrix. At corresponding points $E^{\mathrm{\prime }}(y^{\mathrm{\prime }},t)=QE(y,t)$ and $B^{\mathrm{\prime }}(y^{\mathrm{\prime }},t)=(\det Q)QB(y,t)$. Thus polar and axial components agree in their transformation rule when $\det Q=1$ and differ by a sign when $\det Q=-1$. These rules specify the orientation convention; the adopted inertial descriptions have positive orientation, and no physical orientation-reversing change of description is adopted here.

(b) The fields are defined by their values on $\mathrm{\Omega }\times I$, and no value at a point of singular source support is asserted.

(c) $E$ carries V m${}^{-1}$ and $B$ carries T.

(d) The identification of the pair $(E,B)$ with the components of an alternating two-form, including its transformation behaviour, is made in the framework's electromagnetic two-form definition on this page and is not presupposed here.