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Let $M$ be a [[Differentiable manifold|differentiable manifold]], and let $F(M)$ be the algebra of smooth real-valued functions on it. A tangent vector to $M$ at $m\in M$ is an $\mathbb{R}$-linear mapping $v:F(M)\rightarrow \mathbb{R}$ such that

\begin{equation}\tag{a1}
v(fg)=f(m)v(g)+g(m)v(f).
\end{equation}

For this definition one can equally well (in fact, better) use the ring of germs of smooth functions $F(M,m)$ on $M$ at $m$.

The tangent vectors to $M$ at $m\in M$ form a vector space over $\mathbb{R}$ of dimension $n=\dim (M)$. It is denoted by $T_m M$.

Let $\phi :U\rightarrow \mathbb{R}^n$, $m\mapsto (x_1(m),\dots ,x_n(m))$, where $(x_1,\dots ,x_n)$ is a system of coordinates on $M$ near $m$. The $i$-th partial derivative at $m$ with respect to $\phi$ is the tangent vector

\begin{equation*}
(D_{x_i})(m)(f)=\left.\frac{\partial (f\phi^{-1})}{\partial x_i}\right|_{\phi(m)},
\end{equation*}

where the right hand-side is the usual partial derivative of the function $f\phi^{-1}:\mathbb{R}^n\rightarrow \mathbb{R}$ in the variables $x_1,\dots ,x_n$, at the point $\phi(m)\in\mathbb{R}^n$. One has $D_{x_i}(m)(x_j)=\delta_{ij}$ (the Kronecker delta) and the $D_{x_i}(m)$ form a basis for $T_m M$.

This basis for $T_m M$ determined by the coordinate system $(x_1,\dots ,x_n)$ is often denoted by $\{\partial/\partial x_1,\dots ,\partial/\partial x_n\}$.

A cotangent vector at $m\in M$ is an $\mathbb{R}$-linear mapping $T_m M\rightarrow \mathbb{R}$ such that the cotangent space $T_m^* M$ at $m\in M$ is the dual vector space to $T_m M$. The dual basis to $(\partial/\partial x_1,\dots ,\partial/\partial x_n)$ is denoted by $dx_1,\dots ,dx_n$. One has

\begin{equation*}
dx_i(v)=v(x_i),\qquad v\in T_m M.
\end{equation*}

The disjoint union $TM$ of the tangent spaces $T_m M$, $m\in M$, together with the projection $\pi :TM\rightarrow M$, $\pi(v)=m$ if $v\in T_m M$, can be given the structure of a differentiable [[Vector bundle|vector bundle]], the [[Tangent bundle|tangent bundle]].

Similarly, the cotangent spaces $T_m^* M$ form a vector bundle $T^*M$ dual to $TM$, called the cotangent bundle. The sections of $TM$ are the vector fields on $M$, the sections of $T^*M$ are differentiable $1$-forms on $M$.

Let $\alpha: M\rightarrow N$ be a mapping of differentiable manifolds and let $\alpha^*:F(N)\rightarrow F(M)$ be the induced mapping $g\mapsto g\alpha$. For a tangent vector $v:F(M)\rightarrow \mathbb{R}$ at $m$, composition with $\alpha^*$ gives an $\mathbb{R}$-linear mapping $v\alpha^*:F(N)\rightarrow\mathbb{R}$ which is a tangent vector to $N$ at $\alpha(m)$. This defines a homomorphism of vector spaces $T\alpha(m):T_m M\rightarrow T_{\alpha(m)}N$ and a vector bundle morphism $T\alpha:TM\rightarrow TN$.

In case $M=\mathbb{R}^n$ and $N=\mathbb{R}^m$ with global coordinates $x_1,\dots ,x_n$ and $y_1,\dots ,y_m$, respectively, $\alpha:\mathbb{R}^n\rightarrow \mathbb{R}^m$ is given by $m$ differentiable functions and at each $x\in \mathbb{R^n}$,

\begin{equation*}
T\alpha(x)(\frac{\partial}{\partial x_i})=\frac{\partial y_1}{\partial x_i}(x)\frac{\partial}{\partial y_1}+\dots+\frac{\partial y_m}{\partial x_i}(x)\frac{\partial}{\partial y_m},
\end{equation*}

so that the matrix of $T\alpha(x):T_x\mathbb{R}^n\rightarrow T_{\alpha(x)}\mathbb{R}^m$ with respect to the basis $\partial/\partial x_1,\dots,\partial/\partial x_n$ of $T_x\mathbb{R}^n$ and the basis $\partial/\partial y_1,\dots,\partial/\partial y_m$ of $T_{\alpha(x)}\mathbb{R}^m$ is given by the [[Jacobi matrix|Jacobi matrix]] of $\alpha$ at $x$.

Now, let $M\subset\mathbb{R}^r$ be an embedded manifold. Let $c:\mathbb{R}\rightarrow M\subset\mathbb{R}^n$, $t\mapsto c(t)=(c_1(t),\dots ,c_n(t))$ be a smooth curve in $M$, $c(0)=m$. Then

\begin{equation}\tag{a2}
Tc(0)(\frac{\partial}{\partial t})=\frac{\partial c_1}{\partial t}(0)\frac{\partial}{\partial y_1}+\dots+\frac{\partial c_r}{\partial t}(0)\frac{\partial}{\partial y_r}.
\end{equation}

All tangent vectors in $T_m M\subset T_m\mathbb{R}^r$ arise in this way. Identifying the vector (a2) with the $r$-vector $((\partial c_1/\partial t)(0),\dots,(\partial c_r/\partial t)(0))$, viewed as a directed line segment starting in $m\in M\subset \mathbb{R}^r$, one recovers the intuitive picture of the tangent space $T_m M$ as the $m$-plane in $\mathbb{R}^r$ tangent to $M$ in $m$.

A vector field on a manifold $M$ can be defined as a derivation (cf. [[Derivation in a ring|Derivation in a ring]]) in the $\mathbb{R}$-algebra $F(M)$, $X:F(M)\rightarrow F(M)$. Composition with the evaluation mappings $f\mapsto f(m)$, $m\in M$, yields a family of tangent vectors $X_m\in T_m M$, so that $X$ "becomes" a section of the [[Tangent bundle|tangent bundle]]. Given local coordinates $x_1,\dots,x_n$, $X$ can locally be written as

\begin{equation*}
X=\alpha_1(x)\frac{\partial}{\partial x_1}+\dots+\alpha_n(x)\frac{\partial}{\partial x_n},
\end{equation*}

and if a function $f$ in local coordinates is given by $f(m)=\tilde{f}(x_1(m),\dots,x_n(m))$, then $Xf$ is the function given in local coordinates by the expression

\begin{equation*}
\alpha_1(x)\frac{\partial\tilde{f}}{\partial x_1}+\dots+\alpha_n(x)\frac{\partial\tilde{f}}{\partial x_n},
\end{equation*}

showing once more the convenience of the " $\partial/\partial x_i$" notation for tangent vectors. (Of course, in practice one uses a bit more abuse of notation and writes $f$ instead of $\tilde{f}$.)

Let $\mathcal{E}(m)$ be the $\mathbb{R}$-algebra of germs of smooth functions at $m\in M$ (cf. [[Germ|Germ]]). Let $\mathrm{m}\subset\mathcal{E}$ be the ideal of germs that vanish at zero, and $\mathrm{m}^2$ the ideal generated by all products $fg$ for $f,g\in\mathrm{m}$. If $x_1,\dots,x_n$ are local coordinates at $m$ so that $x(m)=0$, $\mathrm{m}$ is generated as an ideal in $\mathcal{E}$ by $x_1,\dots,x_n\in\mathrm{m}$, and $\mathrm{m}^2$ by the $x_ix_j$, $i,j=1,\dots,n$. In fact, the quotient ring $\mathcal{E}/\mathrm{m}^\infty$ is the power series ring in $n$ variables over $\mathbb{R}$. Here $\mathrm{m}^\infty=\cap_r\mathrm{m}^r$ is the ideal of flat function germs. (A smooth function is flat at a point if it vanishes there with all its derivatives (an example is $\exp(-x^{-2})$ at $0\in\mathbb{R}$); the "Taylor expansion mapping" $\mathcal{E}\rightarrow \mathbb{R}[[x_1,\dots,x_n]]$ is surjective, a very special consequence of the [[Whitney extension theorem|Whitney extension theorem]].)

Now, let $v\in T_mM$ be a tangent vector of $M$ at $m$. Then $v(\mathrm{const})=0$ by (a1) for all constant functions in $\mathcal{E}$. Also $v(\mathrm{m}^2)=0$, again by (a1). Thus, each $v\in T_mM$ defines an element in $\mathrm{Hom}_\mathbb{R}(\mathrm{m}/\mathrm{m}^2,\mathbb{R})$, which is of dimension $n=\dim M$ because $\mathrm{m}/\mathrm{m}^2$ has dimension $n$ (and that element uniquely determines $v$). Moreover, the tangent vectors $\partial/\partial x_1,\dots,\partial/\partial x_n$ clearly define $n$ linearly independent elements in $\mathrm{Hom}_\mathbb{R}(\mathrm{m}/\mathrm{m}^2,\mathbb{R})$ (because $(\partial/\partial x_i)(x_j)=\delta_{ij}$). Thus,

\begin{equation*}
T_mM\simeq\mathrm{Hom}_\mathbb{R}(\mathrm{m}/\mathrm{m}^2,\mathbb{R}),
\end{equation*}

the dual space of $\mathrm{m}/\mathrm{m}^2$. This point of view is more generally applicable and serves as the definition of tangent space in analytic and algebraic geometry, cf. [[Analytic space|Analytic space]]; [[Zariski tangent space|Zariski tangent space]].

====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top"> R. Hermann, "Geometry, physics, and systems" , M. Dekker (1973) {{MR|0494183}} {{ZBL|0285.58001}} </TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top"> R.L. Bishop, R.J. Crittenden, "Geometry of manifolds" , Acad. Press (1964) {{MR|0169148}} {{ZBL|0132.16003}} </TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top"> M.W. Hirsch, "Differential topology" , Springer (1976) pp. 4, 78 {{MR|0448362}} {{ZBL|0356.57001}} </TD></TR><TR><TD valign="top">[a4]</TD> <TD valign="top"> S.P. Novikov, A.T. Fomenko, "Basic elements of differential geometry and topology" , Kluwer (1991) (Translated from Russian) {{MR|1135798}} {{ZBL|0711.53001}} </TD></TR><TR><TD valign="top">[a5]</TD> <TD valign="top"> Yu. Borisovich, N. Bliznyakov, Ya. Izrailevich, T. Fomenko, "Introduction to topology" , Kluwer (1993) (Translated from Russian) {{MR|1450091}} {{MR|0824983}} {{MR|0591670}} {{ZBL|0836.57001}} {{ZBL|0834.57001}} {{ZBL|0478.57001}} </TD></TR></table>