> [!definition] > > Let $\mathbb{S}^1$ be the [[Circle|circle]], $\mathbb{T}^n = (\mathbb{S}^1)^n$ is the $n$-dimensional **torus**. > [!theorem] > > Let $\mathbb{T}^n$ be the $n$-dimensional torus. Let $p \in \mathbb{T}^n$ and let $U_i \in \cn^o(p_i)$ be a [[Connected|connected]] [[Neighbourhood|neighbourhood]] of $p$ in the $i$-th component. Let $\theta^i: U_i \to \real$ be an angle function, and define > $ > \frac{\partial}{\partial \theta^i}: \mathbb{T}^n \to T\mathbb{T}^n \quad p \mapsto \frac{\partial}{\partial \theta^i}\bigg|_p > $ > then $\frac{\partial}{\partial \theta^i}$ is a globally defined, smooth [[Vector Field|vector field]]. > > *Proof*. Let $\seqf{U_i}$ be a family of connected sets such that $p \in \prod U_i = U$. Let $\seqf{\alpha^i}$ and $\seqf{\beta^i}$ be angle functions defined on $\seqf{U_i}$, then $(U, \alpha = (\alpha^i))$ and $(U, \beta = (\beta^i))$ both form coordinate [[Atlas|charts]] at $p$. As each $U_i$ is connected, the angle maps differ by a constant vector $\delta \in \real^n$. Therefore $D(\alpha \circ \beta^{-1})(p) = \text{Id}$. Hence $\frac{\partial}{\partial \alpha^i}\big|_p = \frac{\partial}{\partial\beta^i}\big|_p$, and $\frac{\partial}{\partial \theta^i}$ is well-defined. > > As for each $p \in \mathbb{T}^n$ we can choose a coordinate chart consisting of angle functions, $\frac{\partial}{\partial \theta^i}$ is globally defined. Since $\frac{\partial}{\partial\theta^i}$ is a coordinate chart at each point, it is a smooth vector field.