notation for basis of tangent space of manifold

[demonelite123]

I sometimes see that the basis vectors of the tangent space of a manifold sometimes denoted as ∂/∂x_i which is the ith basis vector. what i am a little confused about is why is the basis vectors in the tangent space given that notation? is there a specific reason for it?

for example, i know that the basis vectors of the cotangent space of a manifold are denoted by dx_i which can be interpreted as the exterior derivative of the coordinate function f(x1,...,x_n) = x_i. is there something similar that allows one to make sense of the notation ∂/∂x_i?

Thanks.

[WannabeNewton]

The notation can be interpreted as the directional derivative at a point on the image of a curve on the manifold that will, like you said, exist in the tangent space at that point.

[Hurkyl]

I sometimes see that the basis vectors of the tangent space of a manifold sometimes denoted as ∂/∂x_i which is the ith basis vector. what i am a little confused about is why is the basis vectors in the tangent space given that notation? is there a specific reason for it?
One of the ways to concretely represent the tangent space is as a certain space of partial differential operators.

One caveat though. While the cotangent field dx_i depends only on the particular vector x_i, the tangent field \partial/\partial x_i depends both on the basis and the vector.

[demonelite123]

thanks for your replies. another thing is that a differential form in the cotangent space is a linear function that takes a vector in the tangent space and maps it to the reals. then for a vector written in terms of the partial differential operators in the tangent space, applying the differential form to it should produce a real number. how would you take the exterior derivative of a partial differential operator though?

also, since you can apply differential forms to vectors is it true that the partial differential operators (vectors) can be applied to a differential form to get a real number since the two spaces are dual to each other?