In what spaces does the notion of a gradient field make sense?

Question

I am doing highschool calculus, and I have been told a gradient field is a field $F(x_1,\cdots,x_n)=f_1(x_1,\cdots,x_n)e_1+\cdots f_n(x_1,\cdots,x_n)e_n$ where we have $$F(x_1,\cdots,x_n)=\nabla q(x_1,\cdots,x_n).$$

(where $\{e_i\}$ are the standard basis vectors for $\Bbb R^n$)

So I guess this definition definitely has $\Bbb R^n$ in mind? Does this notion hold in other spaces? I get that we can define a vectorbundle on really any topological space, and since this notion is apparently equivalent to the notion of a being a conservative field, which seems like it should be very topological, it seems it should (hold in other spaces)?

Related is my previous question: Understanding the conservative property of a vector field

My previous question being related mainly in the sense that it seems that the force field is a section of a bundle, so then we should be able to give a value to each homotopy class of paths, with specific fixed end points, where I feel this is the real definition of a conservative field.

Answer 1

Without getting into too much detail on the definitions involved, on a differential manifold the generalization of the gradient is more naturally interpreted as a section of the cotangent bundle, also known as the space of 1-forms. In this form, the notation would look like: \begin{equation*} df = \frac{\partial f}{\partial x_1} d x_1 + \cdots + \frac{\partial f}{\partial x_n} d x_n. \end{equation*}

In general, though, there is a distinction between the 1-forms which can be expressed in this way, and the generally larger space of 1-forms $\omega$ which satisfy $d \omega = 0$ - I'm not sure which you take as the definition of a "conservative force field". As an example, the restriction to the unit circle of \begin{equation*} \omega := -y\,dx + x\,dy \end{equation*} satisfies $d\omega = 0$ but there is no global function $\theta$ on the circle such that $\omega = d \theta$. (However, there are some functions on subsets of the circle which satisfy this, such as $\theta := \tan^{-1}(y/x)$.) The study of the difference between the two notions is the start of de Rham cohomology which is closely related to homotopy theory.