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Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092200/t0922001.png" /> be a [[Differentiable manifold|differentiable manifold]], and let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092200/t0922002.png" /> be the algebra of smooth real-valued functions on it. A tangent vector to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092200/t0922003.png" /> at <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092200/t0922004.png" /> is an <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092200/t0922005.png" />-linear mapping <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092200/t0922006.png" /> such that
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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
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092200/t0922007.png" /></td> <td valign="top" style="width:5%;text-align:right;">(a1)</td></tr></table>
+
\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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092200/t0922008.png" /> on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092200/t0922009.png" /> at <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092200/t09220010.png" />.
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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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092200/t09220011.png" /> at <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092200/t09220012.png" /> form a vector space over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092200/t09220013.png" /> of dimension <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092200/t09220014.png" />. It is denoted by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092200/t09220015.png" />.
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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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092200/t09220016.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092200/t09220017.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092200/t09220018.png" /> is a system of coordinates on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092200/t09220019.png" /> near <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092200/t09220020.png" />. The <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092200/t09220021.png" />-th partial derivative at <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092200/t09220022.png" /> with respect to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092200/t09220023.png" /> is the tangent vector
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Let $\phi :U\rightarrow \mathbb{R}$, $m\mapsto (x_1(m),\dots ,x_n(m))$, 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
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092200/t09220024.png" /></td> </tr></table>
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\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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092200/t09220025.png" /> in the variables <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092200/t09220026.png" />, at the point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092200/t09220027.png" />. One has <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092200/t09220028.png" /> (the Kronecker delta) and the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092200/t09220029.png" /> form a basis for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092200/t09220030.png" />.
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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 $\phi(m)$. 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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092200/t09220031.png" /> determined by the coordinate system <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092200/t09220032.png" /> is often denoted by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092200/t09220033.png" />.
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This basis for $T_m M$ determined by the coordinate system $(x_1,\dots ,x_n)$ is often denoted by $\{\partial/\partial x_1,\dots ,\aprtial/\partial x_n\}$.
  
A cotangent vector at <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092200/t09220034.png" /> is an <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092200/t09220035.png" />-linear mapping <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092200/t09220036.png" /> such that the cotangent space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092200/t09220037.png" /> at <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092200/t09220038.png" /> is the dual vector space to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092200/t09220039.png" />. The dual basis to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092200/t09220040.png" /> is denoted by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092200/t09220041.png" />. One has
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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 $38$ is the dual vector space to $T_m M$. The dual basis to $(\partial/\partial x_1,\dots ,\aprtial/\partial x_n)$ is denoted by $dx_1,\dots ,dx_n$. One has
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092200/t09220042.png" /></td> </tr></table>
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\begin{equation}
 +
dx_i(v)=v(x_i),\qquad v\in T_m M.
 +
\end{equation}
  
The disjoint union <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092200/t09220043.png" /> of the tangent spaces <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092200/t09220044.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092200/t09220045.png" />, together with the projection <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092200/t09220046.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092200/t09220047.png" /> if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092200/t09220048.png" />, can be given the structure of a differentiable [[Vector bundle|vector bundle]], the [[Tangent bundle|tangent bundle]].
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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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092200/t09220049.png" /> form a vector bundle <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092200/t09220050.png" /> dual to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092200/t09220051.png" />, called the cotangent bundle. The sections of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092200/t09220052.png" /> are the vector fields on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092200/t09220053.png" />, the sections of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092200/t09220054.png" /> are differentiable <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092200/t09220056.png" />-forms on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092200/t09220057.png" />.
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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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092200/t09220058.png" /> be a mapping of differentiable manifolds and let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092200/t09220059.png" /> be the induced mapping <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092200/t09220060.png" />. For a tangent vector <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092200/t09220061.png" /> at <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092200/t09220062.png" />, composition with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092200/t09220063.png" /> gives an <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092200/t09220064.png" />-linear mapping <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092200/t09220065.png" /> which is a tangent vector to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092200/t09220066.png" /> at <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092200/t09220067.png" />. This defines a homomorphism of vector spaces <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092200/t09220068.png" /> and a vector bundle morphism <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092200/t09220069.png" />.
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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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092200/t09220070.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092200/t09220071.png" /> with global coordinates <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092200/t09220072.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092200/t09220073.png" />, respectively, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092200/t09220074.png" /> is given by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092200/t09220075.png" /> differentiable functions and at each <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092200/t09220076.png" />,
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In case $70$ and $71$ with global coordinates $72$ and $73$, respectively, $74$ is given by $75$ differentiable functions and at each $76$,
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092200/t09220077.png" /></td> </tr></table>
+
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;">$77$</td> </tr></table>
  
so that the matrix of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092200/t09220078.png" /> with respect to the basis <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092200/t09220079.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092200/t09220080.png" /> and the basis <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092200/t09220081.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092200/t09220082.png" /> is given by the [[Jacobi matrix|Jacobi matrix]] of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092200/t09220083.png" /> at <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092200/t09220084.png" />.
+
so that the matrix of $78$ with respect to the basis $79$ of $80$ and the basis $81$ of $82$ is given by the [[Jacobi matrix|Jacobi matrix]] of $83$ at $84$.
  
Now, let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092200/t09220085.png" /> be an imbedded manifold. Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092200/t09220086.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092200/t09220087.png" /> be a smooth curve in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092200/t09220088.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092200/t09220089.png" />. Then
+
Now, let $85$ be an imbedded manifold. Let $86$, $87$ be a smooth curve in $88$, $89$. Then
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092200/t09220090.png" /></td> <td valign="top" style="width:5%;text-align:right;">(a2)</td></tr></table>
+
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;">$90$</td> <td valign="top" style="width:5%;text-align:right;">(a2)</td></tr></table>
  
All tangent vectors in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092200/t09220091.png" /> arise in this way. Identifying the vector (a2) with the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092200/t09220092.png" />-vector <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092200/t09220093.png" />, viewed as a directed line segment starting in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092200/t09220094.png" />, one recovers the intuitive picture of the tangent space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092200/t09220095.png" /> as the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092200/t09220096.png" />-plane in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092200/t09220097.png" /> tangent to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092200/t09220098.png" /> in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092200/t09220099.png" />.
+
All tangent vectors in $91$ arise in this way. Identifying the vector (a2) with the $92$-vector $93$, viewed as a directed line segment starting in $94$, one recovers the intuitive picture of the tangent space $95$ as the $96$-plane in $97$ tangent to $98$ in $99$.
  
A vector field on a manifold <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092200/t092200100.png" /> can be defined as a derivation (cf. [[Derivation in a ring|Derivation in a ring]]) in the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092200/t092200101.png" />-algebra <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092200/t092200102.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092200/t092200103.png" />. Composition with the evaluation mappings <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092200/t092200104.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092200/t092200105.png" />, yields a family of tangent vectors <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092200/t092200106.png" />, so that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092200/t092200107.png" /> "becomes" a section of the [[Tangent bundle|tangent bundle]]. Given local coordinates <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092200/t092200108.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092200/t092200109.png" /> can locally be written as
+
A vector field on a manifold $100$ can be defined as a derivation (cf. [[Derivation in a ring|Derivation in a ring]]) in the $101$-algebra $102$, $103$. Composition with the evaluation mappings $104$, $105$, yields a family of tangent vectors $106$, so that $107$ "becomes" a section of the [[Tangent bundle|tangent bundle]]. Given local coordinates $108$, $109$ can locally be written as
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092200/t092200110.png" /></td> </tr></table>
+
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;">$110$</td> </tr></table>
  
and if a function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092200/t092200111.png" /> in local coordinates is given by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092200/t092200112.png" />, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092200/t092200113.png" /> is the function given in local coordinates by the expression
+
and if a function $111$ in local coordinates is given by $112$, then $113$ is the function given in local coordinates by the expression
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092200/t092200114.png" /></td> </tr></table>
+
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;">$114$</td> </tr></table>
  
showing once more the convenience of the " / x" notation for tangent vectors. (Of course, in practice one uses a bit more abuse of notation and writes <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092200/t092200115.png" /> instead of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092200/t092200116.png" />.)
+
showing once more the convenience of the " / x" notation for tangent vectors. (Of course, in practice one uses a bit more abuse of notation and writes $115$ instead of $116$.)
  
Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092200/t092200117.png" /> be the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092200/t092200118.png" />-algebra of germs of smooth functions at <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092200/t092200119.png" /> (cf. [[Germ|Germ]]). Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092200/t092200120.png" /> be the ideal of germs that vanish at zero, and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092200/t092200121.png" /> the ideal generated by all products <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092200/t092200122.png" /> for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092200/t092200123.png" />. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092200/t092200124.png" /> are local coordinates at <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092200/t092200125.png" /> so that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092200/t092200126.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092200/t092200127.png" /> is generated as an ideal in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092200/t092200128.png" /> by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092200/t092200129.png" />, and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092200/t092200130.png" /> by the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092200/t092200131.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092200/t092200132.png" />. In fact, the quotient ring <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092200/t092200133.png" /> is the power series ring in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092200/t092200134.png" /> variables over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092200/t092200135.png" />. Here <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092200/t092200136.png" /> 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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092200/t092200137.png" /> at <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092200/t092200138.png" />); the "Taylor expansion mapping" <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092200/t092200139.png" /> is surjective, a very special consequence of the [[Whitney extension theorem|Whitney extension theorem]].)
+
Let $117$ be the $118$-algebra of germs of smooth functions at $119$ (cf. [[Germ|Germ]]). Let $120$ be the ideal of germs that vanish at zero, and $121$ the ideal generated by all products $122$ for $123$. If $124$ are local coordinates at $125$ so that $126$, $127$ is generated as an ideal in $128$ by $129$, and $130$ by the $131$, $132$. In fact, the quotient ring $133$ is the power series ring in $134$ variables over $135$. Here $136$ 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 $137$ at $138$); the "Taylor expansion mapping" $139$ is surjective, a very special consequence of the [[Whitney extension theorem|Whitney extension theorem]].)
  
Now, let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092200/t092200140.png" /> be a tangent vector of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092200/t092200141.png" /> at <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092200/t092200142.png" />. Then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092200/t092200143.png" /> by (a1) for all constant functions in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092200/t092200144.png" />. Also <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092200/t092200145.png" />, again by (a1). Thus, each <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092200/t092200146.png" /> defines an element in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092200/t092200147.png" />, which is of dimension <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092200/t092200148.png" /> because <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092200/t092200149.png" /> has dimension <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092200/t092200150.png" /> (and that element uniquely determines <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092200/t092200151.png" />). Moreover, the tangent vectors <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092200/t092200152.png" /> clearly define <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092200/t092200153.png" /> linearly independent elements in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092200/t092200154.png" /> (because <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092200/t092200155.png" />). Thus,
+
Now, let $140$ be a tangent vector of $141$ at $142$. Then $143$ by (a1) for all constant functions in $144$. Also $145$, again by (a1). Thus, each $146$ defines an element in $147$, which is of dimension $148$ because $149$ has dimension $150$ (and that element uniquely determines $151$). Moreover, the tangent vectors $152$ clearly define $153$ linearly independent elements in $154$ (because $155$). Thus,
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092200/t092200156.png" /></td> </tr></table>
+
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;">$156$</td> </tr></table>
  
the dual space of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092200/t092200157.png" />. 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]].
+
the dual space of $157$. 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====
 
====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>
 
<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>

Revision as of 15:30, 21 May 2017

Let $M$ be a 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 $\phi :U\rightarrow \mathbb{R}$, $m\mapsto (x_1(m),\dots ,x_n(m))$, 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 $\phi(m)$. 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 $\{\partial/\partial x_1,\dots ,\aprtial/\partial x_n\}$.

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 $38$ is the dual vector space to $T_m M$. The dual basis to $(\partial/\partial x_1,\dots ,\aprtial/\partial x_n)$ 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, the 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 $70$ and $71$ with global coordinates $72$ and $73$, respectively, $74$ is given by $75$ differentiable functions and at each $76$,

$77$

so that the matrix of $78$ with respect to the basis $79$ of $80$ and the basis $81$ of $82$ is given by the Jacobi matrix of $83$ at $84$.

Now, let $85$ be an imbedded manifold. Let $86$, $87$ be a smooth curve in $88$, $89$. Then

$90$ (a2)

All tangent vectors in $91$ arise in this way. Identifying the vector (a2) with the $92$-vector $93$, viewed as a directed line segment starting in $94$, one recovers the intuitive picture of the tangent space $95$ as the $96$-plane in $97$ tangent to $98$ in $99$.

A vector field on a manifold $100$ can be defined as a derivation (cf. Derivation in a ring) in the $101$-algebra $102$, $103$. Composition with the evaluation mappings $104$, $105$, yields a family of tangent vectors $106$, so that $107$ "becomes" a section of the tangent bundle. Given local coordinates $108$, $109$ can locally be written as

$110$

and if a function $111$ in local coordinates is given by $112$, then $113$ is the function given in local coordinates by the expression

$114$

showing once more the convenience of the " / x" notation for tangent vectors. (Of course, in practice one uses a bit more abuse of notation and writes $115$ instead of $116$.)

Let $117$ be the $118$-algebra of germs of smooth functions at $119$ (cf. Germ). Let $120$ be the ideal of germs that vanish at zero, and $121$ the ideal generated by all products $122$ for $123$. If $124$ are local coordinates at $125$ so that $126$, $127$ is generated as an ideal in $128$ by $129$, and $130$ by the $131$, $132$. In fact, the quotient ring $133$ is the power series ring in $134$ variables over $135$. Here $136$ 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 $137$ at $138$); the "Taylor expansion mapping" $139$ is surjective, a very special consequence of the Whitney extension theorem.)

Now, let $140$ be a tangent vector of $141$ at $142$. Then $143$ by (a1) for all constant functions in $144$. Also $145$, again by (a1). Thus, each $146$ defines an element in $147$, which is of dimension $148$ because $149$ has dimension $150$ (and that element uniquely determines $151$). Moreover, the tangent vectors $152$ clearly define $153$ linearly independent elements in $154$ (because $155$). Thus,

$156$

the dual space of $157$. This point of view is more generally applicable and serves as the definition of tangent space in analytic and algebraic geometry, cf. Analytic space; Zariski tangent space.

References

[a1] R. Hermann, "Geometry, physics, and systems" , M. Dekker (1973) MR0494183 Zbl 0285.58001
[a2] R.L. Bishop, R.J. Crittenden, "Geometry of manifolds" , Acad. Press (1964) MR0169148 Zbl 0132.16003
[a3] M.W. Hirsch, "Differential topology" , Springer (1976) pp. 4, 78 MR0448362 Zbl 0356.57001
[a4] S.P. Novikov, A.T. Fomenko, "Basic elements of differential geometry and topology" , Kluwer (1991) (Translated from Russian) MR1135798 Zbl 0711.53001
[a5] Yu. Borisovich, N. Bliznyakov, Ya. Izrailevich, T. Fomenko, "Introduction to topology" , Kluwer (1993) (Translated from Russian) MR1450091 MR0824983 MR0591670 Zbl 0836.57001 Zbl 0834.57001 Zbl 0478.57001
How to Cite This Entry:
Tangent vector. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Tangent_vector&oldid=23991