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. To name the basic coordinate directions, we take the ordinary [partial derivative](/page/Partial%20Derivative) in a chart and reinterpret it as a derivation on the manifold. This gives the standard local basis of the tangent space. [definition: Coordinate Tangent Vector] Let $M$ be a smooth $n$-manifold, let $p\in M$, and let $(U,\varphi)$ be a chart with $p\in U$ and coordinates $(x_1,\ldots,x_n)$. For each $i\in\{1,\ldots,n\}$, the coordinate tangent vector $\partial_{x_i}|_p\in T_pM$ is the derivation $\partial_{x_i}|_p:C_p^\infty(M)\to\mathbb{R}$ given, after choosing a representative of the germ whose domain is restricted inside $U$ if necessary, by \begin{align*} \partial_{x_i}|_p(f)=\frac{\partial(f\circ\varphi^{-1})}{\partial x_i}(\varphi(p)). \end{align*} [/definition] The notation $\partial_{x_i}|_p$ records both the coordinate function and the base point. Omitting the point is harmless only when the base point is fixed by context. Once coordinate directions have been named, the next question is whether they account for every possible tangent direction. [quotetheorem:3904] The coefficients $v_i$ are not intrinsic numbers. They are the components of $v$ in the chosen coordinate frame, just as the same vector in a [finite-dimensional vector space](/page/Finite-Dimensional%20Vector%20Space) has different coordinate columns in different ordered bases. [example: Polar Coordinates and the Same Plane Direction] Polar angle is not a single smooth coordinate on all of the punctured plane $\mathbb{R}^2_0=\mathbb{R}^2\setminus\{0\}$. Choose a local polar chart on an open sector $U\subset\mathbb{R}^2_0$ where a smooth branch of $\theta$ has been fixed, and write the coordinate map to Cartesian coordinates as \begin{align*} x=r\cos\theta,\qquad y=r\sin\theta. \end{align*} Let $p\in U$ have polar coordinates $(r,\theta)$, with $r>0$. The coordinate vector $\partial_r|_p$ is the velocity obtained by varying $r$ while holding $\theta$ fixed. Since \begin{align*} \frac{\partial x}{\partial r}=\cos\theta \end{align*} and \begin{align*} \frac{\partial y}{\partial r}=\sin\theta, \end{align*} we get \begin{align*} \partial_r|_p=\cos\theta\,\partial_x|_p+\sin\theta\,\partial_y|_p. \end{align*} Similarly, $\partial_\theta|_p$ is the velocity obtained by varying $\theta$ while holding $r$ fixed. Differentiating the Cartesian coordinate functions with respect to $\theta$ gives \begin{align*} \frac{\partial x}{\partial\theta}=\frac{\partial}{\partial\theta}(r\cos\theta)=-r\sin\theta \end{align*} and \begin{align*} \frac{\partial y}{\partial\theta}=\frac{\partial}{\partial\theta}(r\sin\theta)=r\cos\theta. \end{align*} Therefore \begin{align*} \partial_\theta|_p=-r\sin\theta\,\partial_x|_p+r\cos\theta\,\partial_y|_p. \end{align*} Its Euclidean squared length is \begin{align*} (-r\sin\theta)^2+(r\cos\theta)^2=r^2\sin^2\theta+r^2\cos^2\theta=r^2(\sin^2\theta+\cos^2\theta)=r^2. \end{align*} Since $r>0$, its Euclidean length is $r$. Also, \begin{align*} (r\cos\theta,r\sin\theta)\cdot(-r\sin\theta,r\cos\theta)=-r^2\cos\theta\sin\theta+r^2\sin\theta\cos\theta=0, \end{align*} so $\partial_\theta|_p$ is perpendicular to the radius vector and hence tangent to the circle of radius $r$ through $p$. The coordinate symbol $\partial_\theta$ therefore does not mean a unit direction; it means the velocity produced by increasing the coordinate $\theta$ at unit speed. [/example] ### Coordinate Change Coordinate changes are the first place where tangent vectors show their tensorial behaviour. Components transform by the [Jacobian matrix](/page/Jacobian%20Matrix) of the coordinate change, while the vector itself remains the same derivation. [quotetheorem:10129] This formula is the computational heart of coordinate independence. It tells us exactly how to translate a velocity measured in one coordinate grid into a velocity measured in another. ## Differentials and Functoriality ### Pushforward of Velocity A tangent vector is useful because smooth maps carry velocities to velocities. If a particle moves through $M$ and $F:M\to N$ observes that motion in another manifold, then the observed velocity should be tangent to $N$. This is the differential of $F$. The construction again uses functions. To understand the velocity of $F\circ\gamma$ in $N$, test it against smooth functions on $N$ and pull those functions back to $M$. [definition: Differential of a Smooth Map] Let $F:M\to N$ be a smooth map between smooth manifolds, and let $p\in M$. The differential of $F$ at $p$ is the linear map $dF_p:T_pM\to T_{F(p)}N$ defined by \begin{align*} (dF_p(v))(h)=v(h\circ F) \end{align*} for every $v\in T_pM$ and every smooth germ $h\in C_{F(p)}^\infty(N)$. [/definition] This definition is short because composition carries local functions on $N$ back to local functions on $M$. It also makes the chain rule natural: pulling a function back through two maps can be done in one step or two. [quotetheorem:3907] The tangent space construction is therefore functorial: manifolds have tangent spaces, and smooth maps have linear maps between tangent spaces. Differential geometry is full of nonlinear objects, but the first-order shadow of every smooth map is linear. [example: Differential of a Projection] Let $\pi:\mathbb{R}^3\to\mathbb{R}^2$ be $\pi(x_1,x_2,x_3)=(x_1,x_2)$, and fix $p=(p_1,p_2,p_3)$. A tangent vector at $p$ represented by $a=(a_1,a_2,a_3)$ is the velocity of the curve \begin{align*} \gamma(t)=p+ta=(p_1+ta_1,p_2+ta_2,p_3+ta_3). \end{align*} Applying $\pi$ to this curve gives \begin{align*} (\pi\circ\gamma)(t)=\pi(p_1+ta_1,p_2+ta_2,p_3+ta_3)=(p_1+ta_1,p_2+ta_2). \end{align*} Differentiating the two coordinate functions at $t=0$ gives \begin{align*} \frac{d}{dt}\Big|_{t=0}(\pi\circ\gamma)(t)=\left(\frac{d}{dt}\Big|_{t=0}(p_1+ta_1),\frac{d}{dt}\Big|_{t=0}(p_2+ta_2)\right)=(a_1,a_2). \end{align*} Thus, under the standard identifications $T_p\mathbb{R}^3\cong\mathbb{R}^3$ and $T_{\pi(p)}\mathbb{R}^2\cong\mathbb{R}^2$, \begin{align*} d\pi_p(a_1,a_2,a_3)=(a_1,a_2). \end{align*} In particular, \begin{align*} d\pi_p(0,0,1)=(0,0). \end{align*} So the vertical direction $(0,0,1)$ is invisible to the projection at first order: moving only in the $x_3$ direction changes no coordinate seen by $\pi$. [/example] ### Rank and Local Invertibility The kernel and image of a differential measure how a map changes dimension to first order. To discuss immersions, submersions, level sets, and local coordinate changes, we need a name for the dimension of the image of the differential. [definition: Rank of a Smooth Map at a Point] Let $F:M\to N$ be a smooth map between smooth manifolds and let $p\in M$. The rank of $F$ at $p$ is \begin{align*} \operatorname{rank}_pF=\dim\operatorname{Range}(dF_p). \end{align*} [/definition] Rank is local first-order information, so the next question is how much nonlinear geometry it actually controls. It does not say whether $F$ is globally injective or surjective, but the extreme full-rank case should force the map to look like a coordinate change near $p$. The [inverse function theorem](/theorems/51) makes this expectation precise: when the differential has no collapsed direction and no missing target direction, the original map becomes reversible after restricting to sufficiently small neighbourhoods. [quotetheorem:10130] This theorem explains why tangent spaces are not merely notation. Invertibility of the linear approximation forces a nonlinear map to be locally reversible. ## Submanifolds and Level Sets ### Constraints and Kernels For a submanifold sitting inside Euclidean space, the tangent space should agree with the geometric picture of tangent lines and tangent planes. The intrinsic definition recovers that picture without making the embedding part of the definition. The cleanest embedded description comes from constraints. If a submanifold is defined by equations, then tangent vectors are exactly velocities that preserve those equations to first order. [quotetheorem:6837] In the notation of the theorem, $C\subset Q$ is the constraint submanifold. When the ambient manifold is $Q=\mathbb{R}^n$ and $M=C$, let $\iota:M\hookrightarrow\mathbb{R}^n$ denote the inclusion map. Its differential $d\iota_p:T_pM\to T_p\mathbb{R}^n$ realizes an intrinsic tangent vector as an ambient velocity. Under this identification, the statement $d\iota_p(T_pM)=\ker dF_p$ is the mathematical version of differentiating the constraint $F(\gamma(t))=c$: allowed velocities are precisely those that do not change the constraint at first order. [example: Tangent Plane to the Sphere] Let $S^2=\{x\in\mathbb{R}^3:|x|^2=1\}$, and define $F:\mathbb{R}^3\to\mathbb{R}$ by \begin{align*} F(x_1,x_2,x_3)=x_1^2+x_2^2+x_3^2. \end{align*} Since $|x|^2=x_1^2+x_2^2+x_3^2$, the equation $|x|^2=1$ is exactly $F(x)=1$, so \begin{align*} S^2=F^{-1}(1). \end{align*} Fix $p=(p_1,p_2,p_3)\in S^2$ and write $v=(v_1,v_2,v_3)\in T_p\mathbb{R}^3\cong\mathbb{R}^3$. The coordinate partial derivatives of $F$ are \begin{align*} \partial_{x_1}F(x)=2x_1,\qquad \partial_{x_2}F(x)=2x_2,\qquad \partial_{x_3}F(x)=2x_3. \end{align*} Using the coordinate formula for the differential in Euclidean space, \begin{align*} dF_p(v)=\partial_{x_1}F(p)v_1+\partial_{x_2}F(p)v_2+\partial_{x_3}F(p)v_3. \end{align*} Substituting the three partial derivatives gives \begin{align*} dF_p(v)=2p_1v_1+2p_2v_2+2p_3v_3. \end{align*} Factoring out $2$ and using the Euclidean dot product, \begin{align*} dF_p(v)=2(p_1v_1+p_2v_2+p_3v_3)=2p\cdot v. \end{align*} Because $p\in S^2$, we have $p\ne 0$, so the linear functional $v\mapsto 2p\cdot v$ is not the zero map. Thus $1$ is a regular value of $F$ along $S^2$, and by *[Regular Level Set Theorem for Holonomic Constraints](/theorems/6837)*, \begin{align*} T_pS^2=\ker dF_p. \end{align*} Now \begin{align*} v\in\ker dF_p \end{align*} means \begin{align*} dF_p(v)=0. \end{align*} Using the formula above, this is equivalent to \begin{align*} 2p\cdot v=0. \end{align*} Since $2\ne 0$, this is equivalent to \begin{align*} p\cdot v=0. \end{align*} Therefore \begin{align*} T_pS^2=\{v\in\mathbb{R}^3:p\cdot v=0\}. \end{align*} The tangent plane consists exactly of the vectors perpendicular to the radius vector $p$, so it is the plane through the origin orthogonal to $p$. [/example] ### Singular Failure A failure of the rank hypothesis produces singularities rather than manifolds. The next example shows that the kernel formula can give the wrong geometric size when the set is not a regular submanifold. [example: A Cusp Where the Naive Tangent Kernel Is Too Large] Let $F:\mathbb{R}^2\to\mathbb{R}$ be defined by $F(x,y)=y^2-x^3$, and let $C=F^{-1}(0)$. The coordinate partial derivatives are \begin{align*} \partial_xF(x,y)=-3x^2,\qquad \partial_yF(x,y)=2y. \end{align*} Hence for $v=(v_1,v_2)\in T_{(0,0)}\mathbb{R}^2\cong\mathbb{R}^2$, \begin{align*} dF_{(0,0)}(v)=\partial_xF(0,0)v_1+\partial_yF(0,0)v_2. \end{align*} Substituting the partial derivatives at the origin gives \begin{align*} dF_{(0,0)}(v)=0\cdot v_1+0\cdot v_2=0. \end{align*} Therefore every vector in $\mathbb{R}^2$ lies in the kernel: \begin{align*} \ker dF_{(0,0)}=\mathbb{R}^2. \end{align*} This kernel is too large to describe the geometric cusp. The curve $\gamma(t)=(t^2,t^3)$ lies in $C$ because \begin{align*} F(\gamma(t))=F(t^2,t^3)=(t^3)^2-(t^2)^3=t^6-t^6=0. \end{align*} Its velocity is \begin{align*} \gamma'(t)=(2t,3t^2), \end{align*} so \begin{align*} \gamma'(0)=(0,0). \end{align*} For $t\ne0$, the direction from the origin to $\gamma(t)$ is represented by \begin{align*} \frac{(t^2,t^3)}{\sqrt{t^4+t^6}}=\frac{(1,t)}{\sqrt{1+t^2}}. \end{align*} As $t\to0$, this tends to $(1,0)$, so the limiting direction of the cusp is the $x$-axis, not all of $\mathbb{R}^2$. The rank of $dF_{(0,0)}$ is $0$, so $0$ is not a regular value at the origin; the regular-level-set tangent-space theorem therefore does not apply there. [/example] This failure is one reason smooth manifold hypotheses matter. Tangent spaces are linear objects attached to smooth points; singular spaces require tangent cones, Zariski tangent spaces, or other replacements. ## Tangent Bundle and Vector Fields ### The Bundle of All Tangent Spaces A single tangent space describes one point. Geometry usually needs a tangent vector at every point: velocities of a flow, directions of a differential equation, gradients of functions on a Riemannian manifold, or infinitesimal symmetries of a Lie group. The tangent bundle packages all pointwise tangent spaces into one manifold. The construction has to remember the base point. A vector in $T_pM$ and a vector in $T_qM$ live in different vector spaces when $p\ne q$, even if coordinates temporarily identify them with copies of $\mathbb{R}^n$. [definition: Tangent Bundle] Let $M$ be a smooth manifold. The tangent bundle of $M$ is the disjoint union \begin{align*} TM=\bigsqcup_{p\in M}T_pM. \end{align*} It comes with the projection map $\pi:TM\to M$ sending each $v\in T_pM$ to $p$. [/definition] The notation $TM$ hides a great deal of structure. In a chart $(U,\varphi)$, a tangent vector over $p\in U$ has coordinates $(x_1(p),\ldots,x_n(p),v_1,\ldots,v_n)$, so the tangent bundle locally looks like $U\times\mathbb{R}^n$. The obstruction is that different charts identify tangent vectors by different coordinate changes, and those changes must be compatible with the manifold structure on the total space. Without this compatibility, $TM$ would only be a set of vector spaces indexed by points, not a smooth manifold carrying all tangent vectors at once. [quotetheorem:3908] The tangent bundle is the stage on which first-order dynamics live, but the bundle by itself only lists all possible velocities. To specify a differential equation, a flow, or an infinitesimal symmetry on a manifold, we need a rule that selects exactly one allowed velocity at each base point. That rule is a vector field. [definition: Vector Field] Let $M$ be a smooth manifold. A vector field on $M$ is a map $X:M\to TM$ such that $\pi\circ X=\operatorname{id}_M$. A vector field is smooth if $X$ is smooth as a map between smooth manifolds. [/definition] ### Vector Fields as Operators Because $X(p)\in T_pM$, a vector field differentiates functions pointwise. This turns geometric directions into first-order differential operators on functions, and it gives a global version of the derivation definition of tangent vectors. We write $\mathfrak{X}(M)$ for the set of smooth vector fields on the smooth manifold $M$. [quotetheorem:10132] This result is the global analogue of the definition of $T_pM$. At one point, tangent vectors are derivations of germs; over the whole manifold, vector fields are derivations of the algebra of smooth functions. [example: A Rotational Vector Field] On $\mathbb{R}^2$, define the smooth vector field \begin{align*} X(x_1,x_2)=-x_2\,\partial_{x_1}|_{(x_1,x_2)}+x_1\,\partial_{x_2}|_{(x_1,x_2)}. \end{align*} Thus, for $f\in C^\infty(\mathbb{R}^2)$ and $p=(x_1,x_2)$, \begin{align*} (Xf)(p)=X(p)(f)=\left(-x_2\,\partial_{x_1}|_p+x_1\,\partial_{x_2}|_p\right)(f). \end{align*} By linearity of tangent vectors as derivations, \begin{align*} (Xf)(x_1,x_2)=-x_2\,\partial_{x_1}f(x_1,x_2)+x_1\,\partial_{x_2}f(x_1,x_2). \end{align*} An integral curve $\gamma(t)=(x_1(t),x_2(t))$ of $X$ must satisfy \begin{align*} \gamma'(t)=X(\gamma(t))=(-x_2(t),x_1(t)). \end{align*} Equating components gives the system \begin{align*} \dot{x}_1(t)=-x_2(t),\qquad \dot{x}_2(t)=x_1(t). \end{align*} Differentiating the first equation with respect to $t$ and then using the second equation gives \begin{align*} \ddot{x}_1(t)=-\dot{x}_2(t)=-x_1(t). \end{align*} Hence $x_1(t)$ satisfies $\ddot{x}_1+x_1=0$, so for initial data $\gamma(0)=(a,b)$, \begin{align*} x_1(t)=a\cos t-b\sin t. \end{align*} Using $\dot{x}_1=-x_2$, we get \begin{align*} x_2(t)=-\dot{x}_1(t)=a\sin t+b\cos t. \end{align*} Therefore the integral curve through $(a,b)$ is \begin{align*} \gamma(t)=(a\cos t-b\sin t,a\sin t+b\cos t). \end{align*} Its squared distance from the origin is \begin{align*} (a\cos t-b\sin t)^2+(a\sin t+b\cos t)^2=a^2(\cos^2t+\sin^2t)+b^2(\sin^2t+\cos^2t)-2ab\cos t\sin t+2ab\sin t\cos t. \end{align*} Using $\sin^2t+\cos^2t=1$ and cancelling the two mixed terms gives \begin{align*} |\gamma(t)|^2=a^2+b^2. \end{align*} Thus each integral curve stays on the circle centered at the origin through its initial point. Also, at $p=(x_1,x_2)$, \begin{align*} p\cdot X(p)=(x_1,x_2)\cdot(-x_2,x_1)=-x_1x_2+x_2x_1=0. \end{align*} So $X(p)$ is perpendicular to the radius vector $p$, and therefore tangent to the circle centered at the origin through $p$. [/example] Vector fields therefore connect tangent spaces to flows and differential equations. The tangent bundle provides the space of possible states for velocities, while a vector field selects one velocity at each state. ## Cotangent Spaces and Pairing Tangent vectors act on functions, but many geometric quantities naturally eat tangent vectors: differentials of functions, covectors, one-forms, and gradients after choosing a Riemannian metric. To describe them, we take the dual vector space. The cotangent space is not an additional kind of tangent direction. It is the space of linear measurements of tangent directions, so the next problem is to identify measurements that arise directly from the smooth structure, before any metric or [inner product](/page/Inner%20Product) is chosen. [definition: Cotangent Space] Let $M$ be a smooth manifold and let $p\in M$. The cotangent space at $p$, denoted $T_p^*M$, is the dual vector space \begin{align*} T_p^*M=(T_pM)^*. \end{align*} An element of $T_p^*M$ is called a covector at $p$. [/definition] A smooth function produces the first covectors we actually need. If $v\in T_pM$ is a possible velocity, then applying $v$ to $f$ gives the rate at which $f$ changes along that velocity; collecting all those rates into one linear functional is the coordinate-free replacement for the row vector of first partial derivatives. This construction is needed because it records directional change without choosing a metric or identifying covectors with vectors. [definition: Differential of a Function] Let $M$ be a smooth manifold, let $f:M\to\mathbb{R}$ be smooth, and let $p\in M$. The differential of $f$ at $p$ is the covector $df_p\in T_p^*M$ defined by \begin{align*} df_p(v)=v(f) \end{align*} for every $v\in T_pM$. [/definition] In coordinates, this is the familiar first-order expansion. Covectors transform oppositely to tangent vectors, which is why gradients require a metric while differentials do not. The same first-order idea applies to a smooth map whose target is another manifold, where there may be no globally defined coordinate functions to differentiate. The differential must then be characterized by what it does to velocities of curves, since curves are the intrinsic way to test tangent vectors. This raises the basic functorial question for tangent spaces: if a tangent vector is represented by a curve through $p$, what tangent vector should its image under a smooth map represent at the target point? The tangent map answers this by transporting curve velocities through the smooth map, giving the coordinate-free derivative needed for maps between manifolds. [quotetheorem:3906] For the special case of a real-valued function $f:M\to\mathbb{R}$, this curve-based characterization recovers the covector $df_p$: it sends a tangent velocity to the corresponding rate of change of $f$. The gradient $\nabla f$ is extra structure, obtained only after a Riemannian metric identifies tangent and cotangent spaces. [example: Differential Versus Gradient] Let $M=\mathbb{R}^2$ with its [Euclidean metric](/page/Euclidean%20Metric), and let \begin{align*} f(x_1,x_2)=x_1^2+x_1x_2. \end{align*} The coordinate partial derivatives are \begin{align*} \partial_{x_1}f(x_1,x_2)=2x_1+x_2. \end{align*} At $p=(1,2)$ this gives \begin{align*} \partial_{x_1}f(p)=2\cdot 1+2=4. \end{align*} The other coordinate partial derivative is \begin{align*} \partial_{x_2}f(x_1,x_2)=x_1. \end{align*} At $p=(1,2)$ this gives \begin{align*} \partial_{x_2}f(p)=1. \end{align*} Since $dx_1|_p$ and $dx_2|_p$ are dual to $\partial_{x_1}|_p$ and $\partial_{x_2}|_p$, the differential is \begin{align*} df_p=4\,dx_1|_p+1\,dx_2|_p. \end{align*} The Euclidean gradient is the unique tangent vector $\nabla f(p)=A\,\partial_{x_1}|_p+B\,\partial_{x_2}|_p$ such that, for every $v=v_1\partial_{x_1}|_p+v_2\partial_{x_2}|_p$, \begin{align*} df_p(v)=\langle \nabla f(p),v\rangle. \end{align*} Using the [dual basis](/theorems/414) relation, \begin{align*} df_p(v)=4\,dx_1|_p(v)+1\,dx_2|_p(v)=4v_1+v_2. \end{align*} Using the Euclidean metric on the coordinate basis, \begin{align*} \langle A\,\partial_{x_1}|_p+B\,\partial_{x_2}|_p,v_1\partial_{x_1}|_p+v_2\partial_{x_2}|_p\rangle=Av_1+Bv_2. \end{align*} Thus $Av_1+Bv_2=4v_1+v_2$ for all $v_1,v_2\in\mathbb{R}$, so taking $(v_1,v_2)=(1,0)$ gives $A=4$, and taking $(v_1,v_2)=(0,1)$ gives $B=1$. Therefore \begin{align*} \nabla f(p)=4\,\partial_{x_1}|_p+1\,\partial_{x_2}|_p. \end{align*} The covector $df_p$ and the tangent vector $\nabla f(p)$ have the same displayed coefficients here because the Euclidean metric identifies $\partial_{x_i}|_p$ with the dual measurement $dx_i|_p$ in these coordinates. [/example] This distinction becomes essential in Riemannian geometry and complex geometry, where metrics, symplectic forms, and complex structures give different ways to convert between tangent and cotangent data. ## Immersions, Submersions, and Local Models ### Injective Differential The rank of the differential controls the local shape of a map. Some maps preserve all tangent directions, some reach all target directions, and some do both. These cases receive special names because they are the standard local models for submanifolds and coordinate projections. When no nonzero tangent vector is killed, the map behaves like a parametrized submanifold to first order. This is the differential-geometric version of a regular parametrisation. [definition: Immersion] Let $F:M\to N$ be a smooth map between smooth manifolds. The map $F$ is an immersion if, for every $p\in M$, the differential $dF_p:T_pM\to T_{F(p)}N$ is injective. [/definition] Immersions can still cross themselves globally. The tangent condition is local and first-order, so it detects whether the parametrisation has collapsed a direction at the point. [example: A Figure-Eight Immersion] Let $F:\mathbb{R}\to\mathbb{R}^2$ be the smooth map \begin{align*} F(t)=(\sin t,\sin 2t). \end{align*} Differentiating each component gives \begin{align*} F'(t)=\left(\frac{d}{dt}\sin t,\frac{d}{dt}\sin 2t\right)=(\cos t,2\cos 2t). \end{align*} To check that this velocity never vanishes, suppose first that $\cos t=0$. Then $t=\frac{\pi}{2}+k\pi$ for some $k\in\mathbb{Z}$, so \begin{align*} 2t=\pi+2k\pi. \end{align*} Hence \begin{align*} \cos 2t=\cos(\pi+2k\pi)=-1, \end{align*} and therefore \begin{align*} 2\cos 2t=-2\ne0. \end{align*} Thus the two components $\cos t$ and $2\cos 2t$ cannot vanish simultaneously. Under the standard identification $T_t\mathbb{R}\cong\mathbb{R}$, a tangent vector $a\in T_t\mathbb{R}$ is carried by the differential to \begin{align*} dF_t(a)=aF'(t)=a(\cos t,2\cos 2t). \end{align*} If $dF_t(a)=0$, then \begin{align*} a(\cos t,2\cos 2t)=(0,0). \end{align*} Since $F'(t)\ne(0,0)$, at least one of $\cos t$ and $2\cos 2t$ is nonzero, so the corresponding component equation forces $a=0$. Therefore $\ker dF_t=\{0\}$, so $dF_t$ is injective for every $t$. The image may cross itself, but no parametrized branch has zero velocity, which is exactly the first-order condition for $F$ to be an immersion. [/example] ### Surjective Differential The dual local model is a map that reaches every target direction to first order. Such maps locally resemble coordinate projections, and their fibres provide the standard source of embedded submanifolds. [definition: Submersion] Let $F:M\to N$ be a smooth map between smooth manifolds. The map $F$ is a submersion if, for every $p\in M$, the differential $dF_p:T_pM\to T_{F(p)}N$ is surjective. [/definition] Submersions explain why regular level sets are manifolds: if $F$ is a submersion, then the fibres of $F$ have tangent spaces equal to kernels of $dF_p$. When the rank is constant but not necessarily maximal, the same philosophy leads to a unified local normal form. [quotetheorem:10133] The constant rank theorem is the bridge from linear algebra to local geometry. It justifies the idea that tangent maps are first-order models of smooth maps, not merely approximations in coordinates. ## Beyond and Connected Topics Tangent spaces are the entry point to [differential geometry](/page/Cambridge%20III%20Differential%20Geometry). Once tangent spaces are available, vector fields, Lie brackets, differential forms, and flows become intrinsic objects rather than coordinate formulas. The next conceptual step is to study how these tangent-space objects vary from point to point, which leads from $T_pM$ to the tangent bundle $TM$ and to differential equations on manifolds. In [Riemannian geometry](/page/Cambridge%20III%20Riemannian%20Geometry), each tangent space receives an inner product $g_p$. This extra structure turns lengths, angles, gradients, geodesics, curvature, and volume into intrinsic notions on manifolds. The distinction between $df_p\in T_p^*M$ and $\nabla f(p)\in T_pM$ becomes central here: the metric is exactly the extra datum that converts one into the other. In algebraic geometry, tangent spaces have an algebraic analogue: the Zariski tangent space. It detects infinitesimal solutions to polynomial equations and is especially sensitive to singularities, making it a natural continuation from the cusp example above. The background appears in [Cambridge II Algebraic Geometry](/page/Cambridge%20II%20Algebraic%20Geometry), where the tangent-space idea is rebuilt from ideals and local rings rather than smooth curves. In complex geometry, tangent spaces carry compatibility with multiplication by $i$, and holomorphic maps have complex-linear differentials. This leads to holomorphic tangent bundles, Hermitian metrics, curvature of vector bundles, and themes developed in [Several Complex Variables IV: Complex Geometry and Curvature](/page/Several%20Complex%20Variables%20IV%3A%20Complex%20Geometry%20and%20Curvature). Tangent spaces also underlie Lie theory. For a Lie group $G$, the tangent space $T_eG$ at the identity is the [Lie algebra](/page/Lie%20Algebra) $\mathfrak{g}$, and group multiplication transports infinitesimal information across the entire group. This is the point where tangent spaces stop being only local linear approximations and become algebraic objects that remember the symmetry of the manifold. ## References Androma, [Cambridge II Algebraic Geometry](/page/Cambridge%20II%20Algebraic%20Geometry). Androma, [Cambridge III Differential Geometry](/page/Cambridge%20III%20Differential%20Geometry). Androma, [Cambridge III Riemannian Geometry](/page/Cambridge%20III%20Riemannian%20Geometry). Androma, [Several Complex Variables IV: Complex Geometry and Curvature](/page/Several%20Complex%20Variables%20IV%3A%20Complex%20Geometry%20and%20Curvature). John M. Lee, *Introduction to Smooth Manifolds* (2013). John M. Lee, *Riemannian Manifolds: An Introduction to Curvature* (1997). Michael Spivak, *A Comprehensive Introduction to Differential Geometry, Volume I* (1979).