The Gauss linking integral is symmetric in its two loops

Let $\gamma_1,\gamma_2:\mathbb{R}\to S^3$ be disjoint $C^1$ loops and let $N\in S^3$ miss both. Then

$\mathrm{Gauss}_N(\gamma_1,\gamma_2)$ denotes gaussLinkingIntegral N γ₁ γ₂, the Gauss double integral $\frac1{4\pi}\int_0^1\!\int_0^1\frac{\operatorname{vol}(A'(s),B'(t),A(s)-B(t))}{|A(s)-B(t)|^3}\,dt\,ds$ of the stereographic images $A=\sigma_N\circ\gamma_1$, $B=\sigma_N\circ\gamma_2$ from the pole $N$.

The integrand is symmetric under exchanging the loops: $\operatorname{vol}(b,a,-u)=\operatorname{vol}(a,b,u)$ (one row swap and one sign change) and $|-u|=|u|$. The integrand is continuous on $[0,1]^2$ because the loops are disjoint, so Fubini's theorem exchanges the order of integration. This is the symmetry $\operatorname{lk}(K_1,K_2)=\operatorname{lk}(K_2,K_1)$ of the linking number in $S^3$.