= Tangent space
{title2=$T_pM$}
{wiki}
At a fixed point $p$ of an $n$-dimensional <smooth manifold> $M$ without boundary, the <tangent space> $T_pM$ is the real <vector space> of all <tangent vectors> at $p$. In <manifold charts> $x=(x^1,\ldots,x^n)$ and $y=(y^1,\ldots,y^n)$, a <tangent vector> has components
$$
v=\sum_{i=1}^n v_x^i\left.\frac{\partial}{\partial x^i}\right|_p,\qquad
v_y^a=\sum_{i=1}^n\left.\frac{\partial y^a}{\partial x^i}\right|_p v_x^i.
$$
The <coordinate basis> transforms inversely, so the <tangent vector> is unchanged. This invertible <Jacobian matrix> also preserves <vector addition> and <scalar multiplication>, giving a <vector space> of <dimension> $n$ independently of coordinates. https://www.math.toronto.edu/mein/teaching/LectureNotes/Man.pdf[Eckhard Meinrenken, Differentiable Manifolds, §1.5: tangent spaces and coordinate bases]{ref}
For example, at $(x,y)=(1,0)$ in $\mathbb R^2$, use the alternative <manifold chart> $u=x$, $w=y+x^2$. The <Jacobian matrix> sends $(v_x,v_y)=(2,3)$ to $(v_u,v_w)=(2,7)$. Indeed, the <smooth curve> $(x(t),y(t))=(1+2t,3t)$ gives $w(t)=1+7t+4t^2$. Conversely, $\partial_u|_p=\partial_x|_p-2\partial_y|_p$ and $\partial_w|_p=\partial_y|_p$, so $2\partial_u|_p+7\partial_w|_p=2\partial_x|_p+3\partial_y|_p$. The numbers change; their action on every <smooth function> does not.
For an <embedded submanifold> $M\subseteq\mathbb R^N$, the <tangent space> can be identified with the <vector subspace>
$$
T_pM=\{\gamma'(0):\gamma:(-\varepsilon,\varepsilon)\to M\text{ is smooth},\ \gamma(0)=p\}.
$$
A proof outline explains why every coordinate vector is possible: in a <manifold chart> $x$, lift the short straight path $x(p)+ta$ using $x^{-1}$. Its <tangent vector> has components $a$. For a regular local parametrization $X:U\subseteq\mathbb R^n\to M$, the same <chain rule> gives $T_pM=\operatorname{im}DX_q$, where $X(q)=p$. This is the <tangent space from a local parametrization>.
For the <sphere> $S_R^2=\{p\in\mathbb R^3:p\cdot p=R^2\}$, differentiating that equation gives $p\cdot v=0$. Conversely, for any nonzero $v$ with $p\cdot v=0$, the <smooth curve>
$$
\gamma_v(t)=p\cos\left(\frac{\|v\|t}{R}\right)+R\frac{v}{\|v\|}\sin\left(\frac{\|v\|t}{R}\right)
$$
lies on the <sphere> and has $\gamma_v'(0)=v$; use a constant <smooth curve> for $v=0$. Thus $T_pS_R^2=\{v:p\cdot v=0\}$, not merely a subset of that plane. At $p=(0,0,R)$ this <vector subspace> is $v_z=0$, while the affine <tangent plane> drawn through $p$ is $z=R$.
This also explains a <mechanical constraint> without requiring a force calculation. In the ideal mathematical model of a particle confined to a stationary <sphere>, its allowed <velocities> at $p$ are exactly $T_pS_R^2$: the radial component must vanish, but either independent tangential direction is available. For a general stationary <holonomic constraint> $F(q)=0$ with independent equations, the <regular level set theorem> gives $T_qM=\ker DF_q$. These are instantaneous kinematic possibilities; they do not determine the actual subsequent motion. On a general <smooth manifold>, <tangent spaces> at different base points cannot be identified canonically; an <affine connection> supplies <parallel transport> when such comparisons are needed.
Back to article page
