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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

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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.

You can also find many questions on this site that explain why we define tangent vectors this way. See, for example, "Why do we think of a vector as being the same as a differential operator?"

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    $\begingroup$ Why the downvote? The OP asked for a book, and I gave one of the standard references. $\endgroup$ – symplectomorphic Mar 11 '16 at 17:39
  • $\begingroup$ But in $\mathbb{R}^3$,$ \dfrac{\partial}{\partial x_i} =e_i$, how can i get it? $\endgroup$ – mnmn1993 Mar 11 '16 at 17:59
  • $\begingroup$ @mnmn1993: I have no idea what "how can I get it?" means. $\endgroup$ – symplectomorphic Mar 11 '16 at 18:01
  • $\begingroup$ In $\mathbb{R}^3$,why $\dfrac{\partial}{\partial x_i} =e_i$? $\endgroup$ – mnmn1993 Mar 11 '16 at 18:02
  • $\begingroup$ @mnmn1993: go read a differential geometry book, or the link in my answer. $\endgroup$ – symplectomorphic Mar 11 '16 at 18:03
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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 $\Bbb R^3$.

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