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I have just learned tangent space in smooth Manifold.It is defined in this manner....Let $M$ be a smooth Manifold,let $\Gamma$$(p)$ be the collection of all smooth curve on $M$ to the point $p$$\in$$M$ such that $d$($\gamma$$(t)$)$/dt$ $=$$d$($\tau$(t))/$dt$, at $t=0$.$\gamma$,$\tau$$\in$$\Gamma$$(p)$.Under these condition we define a relation $\sim$.It is easy to verify that $\sim$ is an equivalence relation and each of the equivalence classes are called tangent vector at $p$.But If I focus in particularly Euclidean Space how can I relate it with the Tangent space in Euclidean space,By Tangent Space on Euclidean Space I know the vector space that is generated by the partial derivaties w.r.t $x_1$,$x_2$,..,$x_n$ where $x_1,x_2,...,x_n$ are coordinates in $R^n$.How can I compare these two tangent space in similar way?Does the tangent space at $p$$\in$ $M$ represents a plane of dimension $n$ in $n$ dimensional Manifold $M$?

I have similar problem in understanding vector field.By Vector Field we mean a section of vector Bundle on a smooth Manifold.But in Euclidean Space by Vector field we mean a mapping $X(p)->(p,x(p))$$p$$\in$$R^n$,,I think there need to be a similarity in the definition of Vector field Euclidean Space and manifold as smooth Manifolds are the general form of Euclidean Spaces,But I can't view that similarity!

How does these two definition in Manifold are generalized from their definition in Euclidean space?

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Indeed, for a point $p\in\mathbb{R}^n$, the tangent space $T_p\mathbb{R}^n$ is spanned by the partial derivatives $\frac{\partial}{\partial x^i},\;i=1,\ldots,n.$ As in the general case, every such tangent vector is represented by an equivalence class of curves - the vector $v=\sum v^i\frac{\partial}{\partial x^i}$ corresponds to all the curves $\gamma$ with $\gamma(0)=p$ and $\dot{\gamma}(0)=v.$ The simplest curve representing the above $v$ is the linear one: $$\gamma(t)=p+t\sum v^ie_i.$$ The above explanation shows that for every $p\in\mathbb{R}^n$ we have a canonical isomorphism $$T_p\mathbb{R}^n\cong\mathbb{R}^n,$$namely, $$\frac{\partial}{\partial x^i}\mapsto e_i$$(this happens only in Euclidean space). This isomorphism makes it possible to think of a vector field on $\mathbb{R}^n$ as a function $X:\mathbb{R}^n\to\mathbb{R}^n,$ especially for people who are not familiar with the language of differential geometry.

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