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An Abelian gauge transformation $A_a\mapsto A_a+{\partial }_a\lambda$ changes $\partial \cdot A$ by $\square \lambda$. Consequently gauge fixing by a quadratic function of $\partial \cdot A$ retains transformations satisfying $\square \lambda =0$, subject to the same boundary conditions. Eliminating those residual transformations is part of specifying an inverse with a particular Green function prescription.

For a planar magnetic field written as $\mathbf{B}=\nabla \times (A\hat{\mathbf{z}})$ and a circular velocity $\mathbf{u}=s\Omega (s)\hat{\mathbf{\varphi }}$, the resistive induction equation reduces, after choosing the additive gauge transformation, to ${\partial }_{t}A+\Omega {\partial }_{\varphi }A=\eta {\nabla }^{2}A$. A Fourier mode $A=a(s,t)e^{im[\varphi -\Omega (s)t]}$ therefore obeys the equation in the title. Indeed, each radial derivative acting on the exponential replaces ${\partial }_s$ by ${\partial }_s-imt{\Omega }^{\prime }$, while the azimuthal Laplacian contributes $-m^2/s^2$. At zero magnetic diffusivity, $a$ is constant in time, but the physical magnetic field can still be amplified algebraically by differential rotation.

Let $k^i$ generate a holomorphic gauge transformation, with $k^i{\partial }_{i}W=-{\kappa }^{2}rW$ and moment map $\mathcal{P}=i(k^i{K}_i-r)$. Substitution of the Kähler covariant derivative of a superpotential gives the displayed identity. Consequently, a finite point with $W\ne 0$ and all $D_{i}W=0$ also has $\mathcal{P}=0$. Pure D-term breaking with vanishing chiral auxiliary fields can therefore evade this argument only where $W=0$ or outside its hypotheses. In conventional two-derivative matter-coupled supergravity, a constant Fayet–Iliopoulos term gauges an R-symmetry and constrains the transformation of $W$; a neutral superpotential and an independently assigned nonzero constant shift are not generally consistent. The gauge-covariance restriction is derived in Van Proeyen's discussion of Fayet–Iliopoulos terms and R-symmetry.

The vector Dirac current has an even temporal component and odd spatial components under time-reversal symmetry. Invariance of its real minimal coupling therefore gives the displayed transformation of the gauge field, up to a gauge transformation. Differentiation at $x_T=(-x^0,\mathbf{x})$ yields $F_{\mu \nu }(x)\mapsto -t_{\mu }{t}_{\nu }{F}_{\mu \nu }(x_T)$, with $t=(1,-1,-1,-1)$. Thus the electric field is even and the magnetic field is odd.

A connection on a principal circle bundle has local real one-form representatives $A$ transforming by $A\mapsto A+d\chi$. Its curvature is the globally defined two-form $F=dA$. In unit-charge normalization its holonomy around a closed loop is $e^{i\oint A}$, and curvature periods are integer multiples of $2\pi$ on closed surfaces. A circulation representative is defined modulo $2\pi$ under large gauge transformations.

An einbein specifies a one-dimensional metric tensor along a worldline. In the relativistic particle phase-space action it is the Lagrange multiplier for the mass-shell condition and transforms as $\delta e=\dot{ϵ}$ under the corresponding canonical gauge transformation. After eliminating the momentum, its geometric interpretation depends on the normalization of the particle action.