Exponential maps of $\mathbb{R}^n$ and $T^n$
Solution 1:
See $S^1$ as $\{z\in\mathbb{C}\,|\,|z|=1\}$. If $X=(t_1,\ldots,t_n)$, consider the map $\gamma\colon\mathbb{R}\longrightarrow\left(S^1\right)^n$ defined by $\gamma(x)=\bigl(e^{ixt_1},e^{ixt_2},\ldots,e^{ixt_n}\bigr)$. It is clear that $\gamma$ is a homomorphism and that $\gamma'(0)=X$. Therefore,$$\exp(X)=\gamma(1)=\bigl(e^{it_1},e^{it_2},\ldots,e^{it_n}\bigr).$$