suppose $\omega$ is a nonempty open set with lipschitz continuous boundary in $\mathbb{r}^n$. there are certain exponents $e\in \mathbb{r}$ and $q\in (1, infty)$ for which $\displaystyle \frac{\partial}{\partial x^j}: w^{e,q}(\omega) \rightarrow w^{e-1,q}(\omega)$ is not a well-defined continuous operator. now suppose $m$ is a compact smooth manifold. in this talk we will try to discuss the following questions: \begin{enumerate} \item how are sobolev spaces of sections of vector bundles on $m$ defined? \item is it possible to extend $d: c^\infty(m)\rightarrow c^\infty(t^{*}m)$ to a continuous linear map from $w^{e,q}(m)$ to $w^{e-1,q}(t^{*}m)$ for all $e\in \mathbb{r}$ and $q\in (1,\infty)$? \item why are we interested in the above question? \end{enumerate}