In classical electrodynamics, given the shape of a wire carrying electric current, it is possible to obtain the magnetic field configuration $\mathbf{B}$ via the Biot-savart law. If the wire is a curve $\gamma$ parametrized as $\mathbf{y}(s)$, where $s$ is the arc-length, then
$$ \mathbf{B}(\mathbf{x}) = \beta \int_\gamma \dfrac{ d\mathbf{y}\times(\mathbf{x} - \mathbf{y}) }{|\mathbf{x} - \mathbf{y}|^3} = \beta \int_\gamma ds \dfrac{ \mathbf{y}'(s) \times(\mathbf{x} - \mathbf{y}(s)) }{|\mathbf{x} - \mathbf{y}(s)|^3} \, , $$
where $\beta $ is just a physical constant proportional to the current in the wire.
Another application of the Biot-Savart law is to find the velocity field $\mathbf{v}$ around a bent vortex line in a fluid, in the approximation of incompressible and irrotational fluid flow (i.e. $\nabla \cdot \mathbf{v} =0$ and $\nabla \times \mathbf{v} =0$ almost everywhere) and very-thin diameter of the vortex core. In fact, by demanding that the vorticity of the fluid is concentrated on the vortex core (i.e. it is distributed as a Dirac delta peaked on the vortex core),
$$ \mathbf{w}(\mathbf{x}) = \nabla \times \mathbf{v}(\mathbf{x})= c \int_\gamma ds \, \mathbf{y}'(s)\, \delta( \mathbf{x} - \mathbf{y}(s)) \, , $$
we have that the Helmholtz decomposition and the fact $\delta(\mathbf{y}-\mathbf{x} ) = -\nabla^2 \, (4 \pi |\mathbf{y}-\mathbf{x}|)^{-1}$ tell us that
$$ \mathbf{v}(\mathbf{x})= \frac{c}{4 \pi} \int_\gamma ds \dfrac{ \mathbf{y}'(s) \times(\mathbf{x} - \mathbf{y}(s)) }{|\mathbf{x} - \mathbf{y}(s)|^3}$$
Again, the constant $c$ is just a physical constant that sets the value of the circulation of the field $\mathbf{v}$ around the vortex.
This is very clear and works in $\mathbb{R}^3$. Imagine now that the wire (or the curve that parametrizes the irrotational vortex) is a curve in the three-dimensional torus $\mathbb{T}^3 = S^1 \times S^1 \times S^1$. How to obtain the equivalent of the Biot-Savart law?
NOTE: we are changing the base manifold from $\mathbb{R}^3$ to $\mathbb{T}^3 $ but the local differential relations should be unchanged (i.e. the definition of the vorticity 2-form as external derivative of the velocity 1-form, or the local form of Maxwell equations $dF = J$). The problem is that the Biot-Savart law is non-local, so it is a global problem that "feels" the topology of the manifold. Maybe in the end the question is related to how the Helmholtz decomposition works on a torus.
from Hot Weekly Questions - Mathematics Stack Exchange
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