Vector differential rigourous definition

Question

I don't understand how vector differential $d\vec{r}$ is defined rigorously. For a differential $dx$, it can be defined rigorously through differential geometry, where $dx$ is interpreted as a covector. However this view doesn't seem to work for $d\vec{r}$. For example, in Cartesian coordinates $d\vec{r} = dx\space \vec{e_x} + dy\space \vec{e_y}$. I can understand $dx + dy$ as a one form, but I don't know how to interpret $dx\space \vec{e_x} + dy\space \vec{e_y}$.

I'm coming from curvilinear coordinates wikipedia article, where $d\vec{r}$ is used extensively to get result that helps prove vector operators in non-Cartesian coordinate system.

Answer 1

I think I now understand it a bit more. To answer this question, I don't $d\mathbf{r}$ is a differential form.

Answer 2

It seems like you know some differential geometry, so I would recommend you read Loomis and Sternberg's book Advanced Calculus. In appendix II of chapter $11$, they introduce the notion of a vector valued differential form. The definition is as follows:

Let $M$ be a differentiable manifold, and $E$ a Banach space over $\Bbb{R}$. An $E$-valued exterior differential form on $M$, of degree $p$, is a function $\omega$ which assigns to each $x \in M$, a function \begin{align} \omega(x): \underbrace{(T_xM) \times \dots \times (T_xM)}_{p \text{ times}} \to E, \end{align} whereby $\omega(x)$ is (continuous) multilinear and alternating.

Typically when we discuss differential forms, we take $E= \Bbb{R}$. Hence, we get the notion of a 1-form like $dx,dy,dz$. If instead we only assume that $E$ is a finite-dimensional real vector space, then by choosing a basis $\{e_1, \dots, e_n\}$ for $E$, we can uniquely write every $E$-valued exterior differential form $\omega$ as \begin{equation} \omega = \sum_{i=1}^n \omega_ie_i \end{equation} where $\omega_i$ are $\Bbb{R}$-valued exterior differential forms (i.e the usual kind which you might be used to). In such a case, we might write the form $\omega$ in vector notation as $(\omega_1, \dots, \omega_n)$ (again, refer to the book for a slightly more detailed explanation).

---

A concrete example can be when $E = \Bbb{R}^2$, and we choose the standard basis $e_1,e_2$. Then, once we have the real-valued 1-forms $dx$ and $dy$, we can define the vector-differential form $d\boldsymbol{r}$ as \begin{equation} d\boldsymbol{r} := dx \cdot e_1 + dy \cdot e_2 = (dx,dy) \end{equation} where the symbols have the meaning as explained above.