Integral Formula for a Generalized Sato–Levine Invariant in Magnetic Hydrodynamics

For a pair of divergence-free vector fields $\mathbf B$ and $\widetilde{\mathbf B}$ respectively localized in two oriented tubes $U$ and $\widetilde U$ in $\mathbb R^3$, we propose a fourth-order integral $W$ and describe the dependence between the integral $W$ and a higher topological invariant $\beta=\beta(l,\widetilde l)$ (namely, the generalized Sato–Levine invariant). The new integral is a generalization of the well-known integral, which was defined earlier for two tubes with zero linking number.