Equilibrium vector potentials in \(\mathbb{R}^ 3\) (Q5917780)

In this comprehensive paper the author studies the magnetic field \(B(x)\) induced by the equilibrium current density \(JdS_x\) on \(\Sigma\) and the equilibrium vector potential \(A(x)\). It is also taken into consideration the total energy \(\nu= \int_{\mathbb{R}^3} |B(x)|^2dv_x\) of the magnetic field \(B(x)\). The vector-functions \(A\) and \(B\) can be expressed as \[ A(x)=(1/4\pi) \cdot\int_\Sigma \bigl(J(y)/ |x-y|\bigr) dS_y \quad \text{for} \quad x\in\mathbb{R}^3 \] and \[ B(x) =\text{rot} A(x)= (1/4\pi)\int_\Sigma \bigl(J(y) \times (x-y)/ |x-y|^3 \bigr)dS_y\quad \text{for} \quad x\in\mathbb{R}^3 \backslash\Sigma \] where \(\Sigma\) is the surface of the domain \(D\in\mathbb{R}^3\) on which the equilibrium current density \(JdS_x\) is placed, \(x,y\in D\) or \(\Sigma\). This problem must be seen in connection with the corresponding one from electrostatics where \(D\) is an electric condenser, \(\Sigma\) its surface which carries the equilibrium charge distribution \(\rho dS_x\) inducing the electric field \(E(x)\) in \(\mathbb{R}^3 \backslash\Sigma\), being identical null in \(D\). In this field of research, the author has published several papers and the latest of these is to complete the present one [J. Math. Kyoto Univ. 36, No. 1, 93-114 (1996)]. Using a large mathematical apparatus, containing partial differential equations (E. Beltrami, A. Weinstein, R. P. Gilbert), real and complex analytical varieties (H. Cartan), potential theory (H. Weil, A. Weinstein) and higher physics studies about electromagnetism (P. P. Feynman at al.), the author analyses the current density surfaces, the vector potentials with zero boundary values, the extremal properties of the solutions and proves the main theorem of the paper: there exist \(q\) linearly independent equilibrium current densities \(\{J_idS_x\}\), \(i=1, \dots,n\) on \(\Sigma\) such that \(J_i[\gamma_j]= \delta_{ij}\) \((1\leq j\leq q)\) (Kronecker's delta). Any equilibrium current density on \(\Sigma\) can be written by a linear combination of \(\{J_idS_x\}\), \(i=1,\dots,q\). The following notations are used: \(\{\gamma_j\}\), \(j=1, \dots,q\) is a base of the 1-dimensional homology group of \(D\), \(J_i[\gamma_j]= \int_Q J(x) \cdot n_xdS_x\) \(\gamma\) denotes a 1-cycle in \(\mathbb{R}^3\) with \(\gamma= \partial Q\). In the final chapter, Examples, the author insists that a formula given in textbooks of electromagnetism for a static magnet field \(B(x)\), in a particular case, is not rigorously correct and he gives its correct form. He also supplies further examples in which the results of the paper are applied to particular cases: The paper is a good example of applied mathematics.