Some very basic geometry problem basis

I am studying Riemannian geometry and have some very basic background for differential geometry.In differential geometry, I only know about thing in $\mathbb{R}^3$ or $\mathbb{R}^2$ , when it comes to some abstract space , my books use $\dfrac{\partial}{\partial x_i}$ and $dx_i$ to denote the basis of tangent space and the cotangent space.

I don't know what is the meaning of $\dfrac{\partial}{\partial x_i}$, because I think it is a differential operator rather than a basis element , what is the meaning of $\dfrac{\partial}{\partial x_i}$ at some point $p$? May I have some reference book or material for this basic thing? Thank you so much

You are correct that $\frac{\partial}{\partial x_i}$ is a differential operator, but in the setting of abstract smooth manifolds, tangent vectors are often defined as differential operators. This is a standard technique you can find in any graduate-level differential geometry book, such as John Lee's Introduction to Smooth Manifolds.
• But in $\mathbb{R}^3$，$\dfrac{\partial}{\partial x_i} =e_i$， how can i get it? – mnmn1993 Mar 11 '16 at 17:59
• In $\mathbb{R}^3$，why $\dfrac{\partial}{\partial x_i} =e_i$? – mnmn1993 Mar 11 '16 at 18:02
At the spaces $\Bbb R^2$ or $\Bbb R^3$ things like $\frac{\partial}{\partial x^i}$ are interpreted as a mechanism to answer how a scalar function $f:\Omega\to\Bbb R$ changes or varies, through the calculation of $\frac{\partial f}{\partial x^i}$ and evaluating at a point $\frac{\partial f}{\partial x^i}(p)$, to get that information. Here $\Omega$ is a subset in $\Bbb R^2$ or $\Bbb R^3$.