Hydrodynamic Biot-Savart kernel ($\mathbf{u}(\mathbf{x})=\frac{1}{4\pi }\int \frac{\mathbf{\omega }(\mathbf{y})\times (\mathbf{x}-\mathbf{y})}{|\mathbf{x}-\mathbf{y}{|}^3}{d}^{3}y$)

In an unbounded incompressible fluid with localized vorticity and no added harmonic background, velocity is reconstructed by $\mathbf{u}(\mathbf{x})=(4\pi {)}^{-1}\int \mathbf{\omega }(\mathbf{y})\times (\mathbf{x}-\mathbf{y})/|\mathbf{x}-\mathbf{y}{|}^3{d}^{3}y$. Its multipole expansion relates hydrodynamic impulse to a dipolar far-field velocity.