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''of a differentiable manifold $M$''

The [[Vector bundle|vector bundle]] $\tau:TM\rightarrow M$, also denoted $\tau (M)$, whose total space $TM$ is given by the union of the tangent spaces $TM|_x$ to $M$ at $x\in M$, consisting of the tangent vectors to $M$, and with projection $\tau$ mapping $TM|_x$ to the point $x$. A section of the tangent bundle $\tau (M)$ is a vector field on $M$ (cf. [[Vector field on a manifold|Vector field on a manifold]]). An atlas on the manifold $TM$ is defined through an atlas of the manifold $M$. The bundle $\tau (M)$ is locally trivial. The transition functions of the tangent bundle are defined by the Jacobi matrices of the transition functions of the atlas of the manifold.

Associated with the tangent bundle is the frame bundle of the manifold $M$, which is a principal bundle. The bundle $\tau^*(M)$ dual to the tangent bundle $\tau (M)$, called the cotangent bundle, consists of the cotangent spaces to $M$. Its sections are the differential or Pfaffian forms (cf. [[Pfaffian form|Pfaffian form]]).

A differentiable mapping $h:M\rightarrow N$ induces a morphism of tangent bundles $\tau(M)\rightarrow \tau(N)$; the corresponding mapping of the total spaces $Th:TM\rightarrow TN$ is called the tangent mapping to $h$ (or differential of $h$). In particular, if $i:M\rightarrow N$ is an immersion (cf. [[Immersion of a manifold|Immersion of a manifold]]), then $\tau(M)$ is a subbundle of the induced vector bundle $i^*\tau(N)$. The quotient bundle $\nu (i)=i^*\tau(N)/\tau(M)$ is called the normal bundle of the immersion. Dually, if $j:M\rightarrow N$ is a [[Submersion|submersion]], then the quotient bundle $\tau(M)/j^*\tau(N)$ is called a subbundle of $j$. If one chooses for $M$ and $N$, $TM$ and $M$ respectively, and $h=\tau:TM\rightarrow M$, then $\tau^*\tau (M)$ is called the tangent bundle of second order.

If $\tau (M)$ is trivial, then $M$ is called a parallelizable manifold.

====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  C. Godbillon,   "Géométrie différentielle et mécanique analytique" , Hermann  (1969)</TD></TR></table>



====Comments====
The tangent mapping (also called differential) $T\alpha (m):T_mM\rightarrow T_{\alpha(m)}N$ induced by a differentiable mapping $\alpha$ is given by

$$T\alpha(m)(v)(g)=v(g\alpha),$$

$g:N\rightarrow \mathbb{R}$, $\alpha:M\rightarrow N$, $v:F(M)\rightarrow \mathbb{R}$, where $F(M)$ is the algebra of smooth functions on $M$ and a [[Tangent vector|tangent vector]] is seen as a special kind of $\mathbb{R}$-linear mapping $F(M)\rightarrow\mathbb{R}$.

In terms of local coordinates and the '$\partial/\partial x_i$-notation' (cf. [[Tangent vector|Tangent vector]]), the matrix of $T\alpha (m)$ is given by the Jacobian matrix of the expression for $\alpha$ in the local coordinates.

There are many notations in use for the differential $T\alpha :TM\rightarrow TN$. Some common ones are: $T\alpha$, $\alpha*$, $J(\alpha)$, $D\alpha$, $d\alpha$. The last one, in case $\alpha$ is a function $\alpha:M\rightarrow \mathbb{R}$,  "sort of agrees"  in both notation and name with $d\alpha$ as the differential $1$-form on $M$ defined by $\alpha$ (cf. [[Differential|Differential]]; [[Differential form|Differential form]]). Using the '$\partial/\partial x_i$ and $dx_i$' notation (cf. [[Tangent vector|Tangent vector]]), the differential $1$-form $d\alpha$ is given in local coordinates by

$$d\alpha=\frac{\partial\alpha}{\partial x_1}dx_1+\dots+\frac{\partial\alpha}{\partial x_n}dx_n$$

(where $\partial\alpha/\partial x_i$ is the result of applying the tangent vector $\partial/\partial x_i$ to $\alpha$). Let $t$ be the coordinate for $\mathbb{R}$. Then $d\alpha:T_mM\rightarrow T_{\alpha(m)}\mathbb{R}$ is given by

$$d\alpha(\frac{\partial}{\partial x_i})=\left(\frac{\partial\alpha}{\partial x_i}\right)\frac{\partial}{\partial t}=$$

$$=\left(\frac{\partial\alpha}{\partial x_1}dx_1+\dots+\frac{\partial\alpha}{\partial x_n}dx_n\right)(\frac{\partial}{\partial x_i})\frac{\partial}{\partial t},$$

because $dx_j(\partial/\partial x_k)=\delta_{jk}$.

The differential $r$-frame bundle over $M$ has as fibre over $m$ the set of all $r$-frames in $T_mM$. (An $r$-frame at $m\in M$ is a set of $r$ independent vectors in $T_mM$. The frame bundle is the differential $n$-frame bundle. A frame on $M$ is a section of the frame bundle and a framed manifold is a manifold provided with a frame.)

====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  M. Spivak,   "Calculus on manifolds" , Benjamin/Cummings  (1965)</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  M.W. Hirsch,   "Differential topology" , Springer  (1976)  pp. Chapt. 5, Sect. 3</TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top">  F. Brickell,   R.S. Clark,   "Differentiable manifolds" , v. Nostrand-Reinhold  (1970)</TD></TR><TR><TD valign="top">[a4]</TD> <TD valign="top">  L. Auslander,   R.E. MacKenzie,   "Introduction to differentiable manifolds" , Dover, reprint  (1977)</TD></TR><TR><TD valign="top">[a5]</TD> <TD valign="top">  R. Hermann,   "Geometry, physics, and systems" , M. Dekker  (1973)</TD></TR><TR><TD valign="top">[a6]</TD> <TD valign="top">  Yu. Borisovich,   N. Bliznyakov,   Ya. Izrailevich,   T. Fomenko,   "Introduction to topology" , Kluwer  (1993)  (Translated from Russian)</TD></TR></table>