Let $M \subseteq \mathbb{R}^n$, $n \in \{2,3\}$, be an open set, and let $u \in C^1([0,T] \times M; \mathbb{R}^n)$ and $p \in C^1([0,T] \times M; \mathbb{R})$ satisfy the incompressible Euler equations \begin{align*} \partial_t u + (u \cdot \nabla)u + \nabla p &= 0, \quad \nabla \cdot u = 0 \end{align*} on $[0,T] \times M$. Let $\Phi: [0,T] \times M \to M$ denote the associated flow map, defined as the unique solution of $\frac{d}{dt}\Phi(t,x) = u(t, \Phi(t,x))$, $\Phi(0,x) = x$. Let $\gamma: [0,1] \to M$ be a $C^1$ parametrisation of a simple closed curve $\Gamma := \gamma([0,1])$ with $\gamma(0) = \gamma(1)$, and define the transported curve \begin{align*} \Gamma(t) &:= \bigl\{ x \in M : \exists\, s \in [0,1] \text{ such that } x = \Phi(t, \gamma(s)) \bigr\}. \end{align*} Then the circulation of $u$ along $\Gamma(t)$, \begin{align*} C(t) &:= \oint_{\Gamma(t)} u \cdot d\ell := \int_0^1 u(t, \Phi(t, \gamma(s))) \cdot \partial_s \Phi(t, \gamma(s))\,d\mathcal{L}^1(s), \end{align*} is conserved in time: $\frac{d}{dt}C(t) = 0$ for all $t \in [0,T]$.