A sphere has no preferred grid. Near the north pole we can draw latitude and longitude, near another point we can use stereographic coordinates, and on an abstract manifold there may be no surrounding Euclidean space at all. Yet differential geometry asks questions that sound linear: what is the velocity of a curve, what is the derivative of a map, what directions are available at a point, and what does it mean for a vector field to vary smoothly? The tangent space is the device that makes these questions intrinsic.

The first difficulty is that a point of a manifold is not a vector. If $M \ne \mathbb{R}^n$, subtracting two nearby points $q-p$ is not defined in any coordinate-free way. A curve through $p$ still has a velocity, but the velocity must be an object attached to $p$, not another point of $M$. Tangent spaces replace forbidden subtraction of points by legitimate first-order action on functions.

[example: Velocity on a Circle] Let $S^1=\{(x,y)\in\mathbb{R}^2:x^2+y^2=1\}$ and let $\gamma:\mathbb{R}\to S^1$ be the smooth curve $\gamma(t)=(\cos t,\sin t)$. Since $\cos^2t+\sin^2t=1$, the image of $\gamma$ lies in $S^1$, and at $t=0$ we have \begin{align*} p=\gamma(0)=(\cos 0,\sin 0)=(1,0). \end{align*} Differentiating the two coordinate functions gives \begin{align*} \gamma'(t)=(-\sin t,\cos t). \end{align*} Therefore \begin{align*} \gamma'(0)=(-\sin 0,\cos 0)=(0,1). \end{align*} This vector is not a point of $S^1$, since $(0,1)\in S^1$ is a different point from $p$; instead it is an element of the vector line through the origin parallel to the affine tangent line to the circle at $p$. If $f\in C^\infty(S^1)$ and $\tilde f$ is a smooth extension to a neighbourhood of $S^1$ in $\mathbb{R}^2$, then $f(\gamma(t))=\tilde f(\cos t,\sin t)$ for all $t$. By the ordinary chain rule in $\mathbb{R}^2$, \begin{align*} \frac{d}{dt}\tilde f(\cos t,\sin t)=\partial_x\tilde f(\cos t,\sin t)(-\sin t)+\partial_y\tilde f(\cos t,\sin t)\cos t. \end{align*} Evaluating at $t=0$ gives \begin{align*} v(f)=\frac{d}{dt}\Big|_{t=0}f(\gamma(t))=\partial_x\tilde f(1,0)\cdot 0+\partial_y\tilde f(1,0)\cdot 1. \end{align*} Thus \begin{align*} v(f)=\partial_y\tilde f(1,0)=\nabla\tilde f(p)\cdot(0,1). \end{align*} The same geometric velocity can therefore be read either as the ambient tangent direction $(0,1)$ at $p$ or as the derivation that sends each smooth function on the circle to its first-order change along $\gamma$. [/example]

This example shows the main principle: tangent vectors are first-order probes. They do not measure the value of a function at $p$; they measure its first-order change along an allowed direction through $p$. The rest of the chapter develops several equivalent ways to make that sentence precise and explains how they interact with maps, coordinates, vector fields, and familiar submanifolds.

## Definition

The most robust definition should not depend on embedding the manifold into Euclidean space. A smooth real-valued function near $p$ is available on every [smooth manifold](/page/Smooth%20Manifold), and directional differentiation of such functions obeys a product rule. The tangent space is built by taking that product rule as the intrinsic signature of first-order motion.

[definition: Tangent Space] Let $M$ be a smooth manifold, let $p\in M$, and let $C_p^\infty(M)$ be the algebra of germs at $p$ of smooth functions $f:U\to\mathbb{R}$ defined on open neighbourhoods $U\subset M$ of $p$. The tangent space to $M$ at $p$, denoted $T_pM$, is the real [vector space](/page/Vector%20Space) of all $\mathbb{R}$-linear maps $v:C_p^\infty(M)\to\mathbb{R}$ such that, for all $f,g\in C_p^\infty(M)$, \begin{align*} v(fg)=f(p)v(g)+g(p)v(f). \end{align*} [/definition]

## Derivations and Curves

The definition above gives the whole space $T_pM$, but the next task is to talk about a single first-order probe without repeatedly unpacking the entire space. A curve velocity, a coordinate direction, and the value of a vector field at $p$ should all be treated as the same kind of object once they act on germs by the Leibniz rule. Naming that object a derivation at $p$ lets later statements focus on the operator itself: its domain, its target, and the product rule that makes it a legitimate infinitesimal direction.

[definition: Derivation at a Point] Let $M$ be a smooth manifold and let $p\in M$. A derivation at $p$ is an $\mathbb{R}$-[linear map](/page/Linear%20Map) $v:C_p^\infty(M)\to\mathbb{R}$ such that, for all $f,g\in C_p^\infty(M)$, \begin{align*} v(fg)=f(p)v(g)+g(p)v(f). \end{align*} [/definition]

This definition is intrinsic, but it can feel indirect at first: it describes tangent vectors by how they act on germs, rather than by drawing arrows or choosing coordinates. One immediate test of the definition is what it does to functions that contain no first-order information. If $c$ denotes the constant germ with value $c$, then the Leibniz rule applied to $1\cdot 1$ gives $v(1)=2v(1)$, hence $v(1)=0$; by linearity, $v(c)=cv(1)=0$. Thus a derivation cannot detect constants, which is exactly what one expects from an infinitesimal direction.

After that sanity check, the next comparison is with curves. A smooth curve through $p$ should produce a derivation by differentiating functions along the curve, and the worked coordinate example below makes this connection explicit in the familiar model case $M=\mathbb{R}^n$. Curves supply intuition and computations; derivations supply an intrinsic definition that survives changes of coordinates and maps between manifolds.

[example: Coordinate Velocity as a Derivation] Let $M=\mathbb{R}^n$, write $p=(p_1,\ldots,p_n)$, and let $a=(a_1,\ldots,a_n)\in\mathbb{R}^n$. For the curve $\gamma(t)=p+ta$, its $i$th coordinate is $\gamma_i(t)=p_i+ta_i$, so $\gamma(0)=p$ and $\gamma_i'(t)=a_i$. For $f\in C^\infty(\mathbb{R}^n)$, define \begin{align*} v_a(f)=\frac{d}{dt}\Big|_{t=0} f(\gamma(t)). \end{align*} By the ordinary chain rule in $\mathbb{R}^n$, \begin{align*} \frac{d}{dt}f(\gamma(t))=\sum_{i=1}^n \partial_{x_i}f(\gamma(t))\,\gamma_i'(t)=\sum_{i=1}^n a_i\,\partial_{x_i}f(p+ta). \end{align*} Evaluating at $t=0$ gives \begin{align*} v_a(f)=\sum_{i=1}^n a_i\,\partial_{x_i}f(p). \end{align*} This operator is a derivation at $p$. Indeed, for $f,g\in C^\infty(\mathbb{R}^n)$, the one-variable product rule gives \begin{align*} v_a(fg)=\frac{d}{dt}\Big|_{t=0}\bigl(f(\gamma(t))g(\gamma(t))\bigr)=v_a(f)g(p)+f(p)v_a(g). \end{align*} Also, for scalars $\lambda,\mu\in\mathbb{R}$, \begin{align*} v_a(\lambda f+\mu g)=\lambda v_a(f)+\mu v_a(g) \end{align*} by linearity of the one-variable derivative. Hence the familiar vector $a$ is represented at $p$ by the differential operator \begin{align*} v_a=\sum_{i=1}^n a_i\,\partial_{x_i}|_p. \end{align*} In Euclidean space this identifies tangent vectors with ordinary vectors, while on a general manifold the same description appears only after choosing local coordinates. [/example]

The algebraic definition also explains why constants have zero derivative. This is not just a pleasant consequence; it is a test that the definition is measuring first-order change rather than absolute function value. Without this fact, tangent vectors would not behave like velocities, because a velocity should not change when we add a locally constant offset to every function being observed.

There is still a compatibility question to settle. We have met tangent vectors as derivations on germs, while geometric intuition often starts from curves passing through the point. The next result identifies these two models: differentiating functions along curves produces exactly the same tangent vectors as the derivation definition, and no tangent vector is missed by doing so.

[quotetheorem:3905]

This is the bridge between the geometric and algebraic languages for tangent space. A smooth curve gives a derivation by measuring the rate of change of every smooth function along the curve, and the theorem says that every derivation at the point arises in this way. Thus one may compute with curves when geometry is visible, or with derivations when functions and coordinates are more convenient, without changing the underlying tangent vector.

## Coordinates and Bases

### Coordinate Directions

Coordinates are not part of the definition of $T_pM$, but they are the main way to compute with it. A coordinate chart turns a piece of $M$ into an open subset of $\mathbb{R}^n$, so every tangent vector should acquire $n$ components. The important point is that the components depend on the chart, while the tangent vector does not.