Level 3B: Tangent Spaces – Derivations and the Differential

The Defect. The picture-book definition of a tangent vector is “an arrow that touches the surface at a point”. On an embedded surface in $\mathbb{R}^3$ that picture works; on an abstract manifold there is no ambient space for the arrow to live in, and the definition collapses. The repair is one of the great relabellings of mathematics: a tangent vector is not an object that points, it is an operator that differentiates. This page builds that operator rigorously, proves it carries exactly the information the arrow carried, and harvests three payoffs: the change-of-basis law, the chain rule on manifolds, and the inverse function theorem on manifolds.

The One-Line Reveal. A tangent vector at $p$ is a linear map that eats smooth functions and returns numbers, satisfying the Leibniz product rule at $p$: everything else (coordinates, the Jacobian, velocities) is the shadow of that single definition under transport by charts.

Scope. From Level 3A: smooth manifolds, charts, admissible atlases, smooth maps and their chart representations, smoothness implies continuity (Proposition 5.1), compositions (Proposition 5.3), the Euclidean derivative with its algebra (Propositions 1.1 and 1.2). Two further inputs are declared, in addition to 3A’s chain rule and inverse function theorem: the fundamental theorem of calculus and differentiation under the integral sign for one-variable integrals of smooth functions (both proved in the Real Analysis track); both are used inside the proof of Hadamard’s lemma, whose statement is given there in full. Everything else is proved here. One word of calibration: from this page on, “smooth” means differentiable of all orders; 3A’s definitions and theorems apply verbatim in the $C^\infty$ category, since the chain rule and inverse function theorem are standardly stated there.

Definition. Let $a \in \mathbb{R}^n$. Two smooth functions defined on (arbitrary open) neighbourhoods of $a$ are equivalent as germs if they agree on some neighbourhood of $a$. A germ at $a$ is an equivalence class; $[f]$ denotes the germ of $f$. Pointwise addition, multiplication, and scalar multiplication of representatives make the set of germs into a commutative ring $C^\infty_a(\mathbb{R}^n)$: if $f_1$ agrees with $f_2$ and $g_1$ with $g_2$ near $a$, then $f_1 + g_1$ agrees with $f_2 + g_2$ and $f_1 g_1$ with $f_2 g_2$ near $a$, so the operations do not depend on the chosen representatives. The value of a germ at $a$ is the common value of its representatives at $a$, written $[f](a)$; it is well defined for the same reason.

Definition. A derivation at $a$ is a map $D: C^\infty_a(\mathbb{R}^n) \to \mathbb{R}$ that is linear and satisfies the Leibniz rule at $a$:

Lemma 1.1 (constants die). $D([c]) = 0$ for every constant germ.

Proof. $D([1]) = D([1] \cdot [1]) = D([1])\,[1](a) + [1](a)\,D([1]) = 2D([1])$, so $D([1]) = 0$; and $[c] = c[1]$ with $D$ linear. $\blacksquare$

Lemma 1.2 (Hadamard’s lemma). Let $f$ be smooth on an open $U \subseteq \mathbb{R}^n$, let $a \in U$, and let $B$ be an open ball around $a$ with $B \subseteq U$. Then there are smooth functions $g_1, \dots, g_n: B \to \mathbb{R}$ with

Proof. For $x \in B$ the segment $a + t(x - a)$, $t \in [0, 1]$, lies in $B$ (balls are convex). Define

First, $g_i(a) = \int_0^1 \partial_i f(a)\, dt = \partial_i f(a)$. Second, the identity: the one-variable function $h(t) = f(a + t(x-a))$ is differentiable on $[0, 1]$ with $h'(t) = \sum_i \partial_i f(a + t(x-a)) (x_i - a_i)$ (the chain rule composed with the affine path, with partials extracted from the Jacobian by 3A’s Proposition 1.1). The fundamental theorem of calculus (the declared input) gives $h(1) - h(0) = \int_0^1 h'(t)\, dt$, that is,

Third, $g_i$ is smooth in $x$: differentiation under the integral sign (the other declared input) gives $\partial_j g_i(x) = \int_0^1 \partial_j \partial_i f(a + t(x-a))\, dt$, and since $f$ is smooth of all orders this proceeds inductively through every order. $\blacksquare$

Theorem 1.3 (derivations are directional differentiation). For every derivation $D$ at $a$ there is a unique vector $v = (v_1, \dots, v_n) \in \mathbb{R}^n$, namely $v_i = D([x_i])$, such that

for every germ $[f]$. Conversely, every $v \in \mathbb{R}^n$ defines a derivation by that formula.

Proof. Well-definedness of the converse first. The number $\partial_i f(a)$ depends only on the germ: two representatives agree on a neighbourhood of $a$, so their $i$-th partials at $a$ agree (partials use values arbitrarily near $a$). Linearity is linearity of partials; the Leibniz rule is the product rule for partials, which follows from 3A’s Propositions 1.1 and 1.2 (the product of differentiable maps is differentiable, so the product’s Jacobian entry is $\partial_i(fg)(a) = \partial_i f(a)\, g(a) + f(a)\, \partial_i g(a)$). Now let $D$ be a derivation. Put $v_i = D([x_i])$. Take a representative $f$ of a germ and restrict attention to a ball $B$ around $a$ inside its domain; Hadamard’s lemma (Lemma 1.2) gives smooth $g_i$ on $B$ with $f = f(a) + \sum_i (x_i - a_i) g_i$ on $B$, so as germs:

Apply $D$ and use linearity, Lemma 1.1, and the Leibniz rule:

The constant term dies (Lemma 1.1); $D([x_i - a_i]) = D([x_i]) - a_i D([1]) = v_i$; and $[x_i - a_i](a) = 0$. So $D([f]) = \sum_i v_i\, \partial_i f(a)$: the formula holds, and it plainly determines $D$ from $v$, giving uniqueness. $\blacksquare$

Notation. The derivation given by $v = e_i$ is written $\frac{\partial}{\partial x_i}\big|_a$: it acts by $\frac{\partial}{\partial x_i}\big|_a [f] = \partial_i f(a)$. Theorem 1.3 says these $n$ operators form a basis of the space of derivations.

Quick check.

1. When are two functions the same germ at a?
2. State the Leibniz rule for a derivation at a.
3. Why does a derivation kill constant germs?
4. State Hadamard’s lemma.
5. Which two declared analysis inputs enter Hadamard’s proof?

Answers: 1. They agree on some neighbourhood of a. 2. D of the product is D(f) g(a) plus f(a) D(g). 3. D(1) = D(1 times 1) = 2 D(1), so D(1) = 0 (Lemma 1.1). 4. f(x) = f(a) + sum (x_i - a_i) g_i(x) with g_i smooth, g_i(a) the i-th partial at a (Lemma 1.2). 5. The fundamental theorem of calculus; differentiation under the integral sign.

Definition. A function $f: M \to \mathbb{R}$ on a smooth manifold is smooth if it is smooth in the sense of Level 3A, Section 5, with $\mathbb{R}$ carrying its standard smooth structure. The germ algebra $C^\infty_p(M)$ at $p \in M$ is the ring of germs at $p$ of smooth functions defined on neighbourhoods of $p$, with pointwise operations (well defined exactly as in Section 1).

Definition. The tangent space of $M$ at $p$ is

the vector space of derivations at $p$, with pointwise operations: $(X + Y)([f]) = X([f]) + Y([f])$ and $(\lambda X)([f]) = \lambda X([f])$. Each $X + Y$ and $\lambda X$ is again a derivation (the Leibniz rule is pointwise), so $T_pM$ is a vector space. Its elements are tangent vectors at $p$.

Theorem 2.1 (charts identify the tangent space with $\mathbb{R}^n$). Let $(U, \varphi)$ be an admissible chart with $p \in U$. Then the transport map

(where $x_1, \dots, x_n$ are the coordinate functions of $\mathbb{R}^n$, so $x_i \circ \varphi$ is a smooth function on $U$) is a linear isomorphism. In particular $\dim T_pM = n$, and $T_pM$ admits the basis

whose transport is the standard basis of $\mathbb{R}^n$.

Proof. Chart maps are smooth. The chart $\varphi: U \to \varphi(U)$ and its inverse $\varphi^{-1}: \varphi(U) \to U$ are smooth maps of manifolds ($U$ and $\varphi(U)$ carry the open-submanifold structures of 3A’s Proposition 2.2, and the chart representation of $\varphi^{-1}$ through the chart $\varphi$ itself is the identity of the open set $\varphi(U)$).

Germ transport is an algebra isomorphism. Composition with $\varphi^{-1}$ carries germs at $p$ to germs at $\varphi(p)$: a smooth $f$ on a neighbourhood of $p$ gives the smooth $f \circ \varphi^{-1}$ on the open neighbourhood $\varphi(U)$ of $\varphi(p)$ (a composition of smooth maps, 3A’s Proposition 5.3); if $f$ and $g$ agree near $p$ then $f \circ \varphi^{-1}$ and $g \circ \varphi^{-1}$ agree near $\varphi(p)$ (continuity of $\varphi^{-1}$), so the transport is well defined on germs; it is an algebra homomorphism ($(fg) \circ \varphi^{-1} = (f \circ \varphi^{-1})(g \circ \varphi^{-1})$ pointwise); and it is bijective with inverse $[h] \mapsto [h \circ \varphi]$: the composites $(h \circ \varphi) \circ \varphi^{-1}$ and $(f \circ \varphi^{-1}) \circ \varphi$ recover $h$ near $\varphi(p)$ and $f$ near $p$ respectively. In the same way, composition with $\varphi$ is an algebra isomorphism $C^\infty_{\varphi(p)}(\mathbb{R}^n) \to C^\infty_p(M)$.

The basis derivations. For $v \in \mathbb{R}^n$ define

Each $X^{\varphi}_v$ is a derivation at $p$: linearity is clear, and the Leibniz rule transports, since $(fg) \circ \varphi^{-1} = (f \circ \varphi^{-1})(g \circ \varphi^{-1})$ and the Euclidean derivation $\sum_i v_i \partial_i$ satisfies the Leibniz rule at $\varphi(p)$ (Theorem 1.3’s converse). The map $v \mapsto X^{\varphi}_v$ is linear, and injective: evaluate on the coordinate germs $[x_j \circ \varphi]$, whose transported representative $(x_j \circ \varphi) \circ \varphi^{-1} = x_j$ has partials $\partial_i x_j = \delta_{ij}$ at $\varphi(p)$, giving $X^{\varphi}_v([x_j \circ \varphi]) = v_j$. So the transport map of the theorem, applied to $X^{\varphi}_v$, returns exactly $v$; and the basis claim holds: the transport of $X^{\varphi}_{e_i}$ is $e_i$.

Surjectivity. Given $X \in T_pM$, define $Y$ on Euclidean germs by $Y([h]) = X([h \circ \varphi])$ (well defined: representatives agreeing near $\varphi(p)$ compose with $\varphi$ to functions agreeing near $p$). $Y$ is a derivation at $\varphi(p)$ (the Leibniz rule transports exactly as above), so Theorem 1.3 supplies $v \in \mathbb{R}^n$ with $v_i = Y([x_i]) = X([x_i \circ \varphi])$ and $Y([h]) = \sum_i v_i\, \partial_i h(\varphi(p))$ for every germ $[h]$. Now for an arbitrary $[f] \in C^\infty_p(M)$, take $h = [f \circ \varphi^{-1}]$, so that $[f] = [h \circ \varphi]$:

So $X = X^{\varphi}_v$: the transport is surjective, hence a linear isomorphism, and $\dim T_pM = n$. $\blacksquare$

Theorem 2.2 (the change-of-basis law). Let $(U, \varphi)$ and $(V, \psi)$ be two admissible charts at $p$. Then

where $y^1, \dots, y^n$ are the coordinates of the $\psi$-chart.

Proof. Both sides are derivations, and a derivation is determined by its values on germs (Theorem 1.3’s uniqueness); so evaluate on an arbitrary $[f] \in C^\infty_p(M)$. The right side gives $\sum_j \partial_i(\psi \circ \varphi^{-1})_j (\varphi(p)) \cdot \partial_j (f \circ \psi^{-1}) (\psi(p))$. By the Euclidean chain rule applied to the smooth composition $(f \circ \psi^{-1}) \circ (\psi \circ \varphi^{-1}) = f \circ \varphi^{-1}$, this equals

which is the left side’s value. $\blacksquare$

This is the classical “tangent vectors transform covariantly” formula, now a theorem rather than a slogan. Its practical content: the basis $\frac{\partial}{\partial x^i}$ changes by the Jacobian of the transition map, which is why the same physical direction acquires different coordinate descriptions in different charts.

Quick check.

1. Define the tangent space at p.
2. What is the chart transport of Theorem 2.1?
3. Write the basis derivation as an operator on germs.
4. How do basis derivations transform between two charts?
5. What is the dimension of the tangent space?

Answers: 1. The derivations of the germ algebra at p. 2. X maps to the list of X applied to the coordinate germs, an isomorphism onto R^n. 3. A germ f maps to the i-th partial of f composed with phi inverse, at phi(p). 4. By the Jacobian matrix of the transition map (Theorem 2.2). 5. n: the transport is a linear isomorphism.

Definition. For a smooth map $f: M \to N$ and $p \in M$, the differential of $f$ at $p$ is

for $[g] \in C^\infty_{f(p)}(N)$.

Proposition 3.1 (the differential is well defined). $[g \circ f]$ is a germ at $f(p)$; the definition does not depend on the representative $g$; and $df_p(X)$ is a derivation. Hence $df_p$ is a linear map between tangent spaces.

Proof. $g$ is defined and smooth on some neighbourhood $W$ of $f(p)$; then $g \circ f$ is defined on $f^{-1}(W)$, which contains a neighbourhood of $p$ (Proposition 5.1 of 3A: $f$ is continuous at $p$), and is smooth there (composition, Proposition 5.3 of 3A): so $[g \circ f]$ is a germ at $p$‘s image point, i.e. at $f(p)$. If $g$ and $g'$ agree near $f(p)$, then $g \circ f$ and $g' \circ f$ agree near $p$ (again continuity of $f$), so the germs agree and $X$ takes the same value. Leibniz for $df_p(X)$ is Leibniz for $X$ applied to $(g_1 g_2) \circ f = (g_1 \circ f)(g_2 \circ f)$. Linearity of $df_p$ is linearity of $X$ pointwise. $\blacksquare$

Theorem 3.2 (the differential in charts is the Jacobian). Let $(U, \varphi)$ and $(V, \psi)$ be charts at $p$ and $f(p)$ with $f(U) \subseteq V$, and let $F = \psi \circ f \circ \varphi^{-1}: \varphi(U) \to \psi(V)$ be the chart representation. For $X = \sum_i v_i \frac{\partial}{\partial x^i} \big|_p$:

the matrix of $df_p$ in the chart bases is the Jacobian matrix of $F$ at $\varphi(p)$.

Proof. Both sides are derivations at $f(p)$; evaluate on the coordinate germs $[y^j \circ \psi]$. The right side contributes its own $j$-th coefficient. The left side:

since $y^j \circ \psi \circ f \circ \varphi^{-1} = y^j \circ F = F_j$. $\blacksquare$

Theorem 3.3 (the chain rule on manifolds). For smooth maps $f: M \to N$ and $g: N \to P$, and $p \in M$:

Proof. Both sides are linear maps $T_pM \to T_{g(f(p))}P$; evaluate on $X \in T_pM$ and an arbitrary germ $[h] \in C^\infty_{g(f(p))}(P)$:

the middle step being the definition of $df_p$ applied to the germ of $h \circ g$ at $f(p)$ (a legitimate germ: $h$ is defined near $g(f(p))$, so $h \circ g$ is defined near $f(p)$ and smooth there). $\blacksquare$

Example (velocities). A smooth curve is a smooth map $\gamma: (a, b) \to M$ from an interval of $\mathbb{R}$. Its velocity at $t$ is $d\gamma_t(\frac{d}{dt}\big|_t)$, where $\frac{d}{dt}\big|_t$ is the standard basis derivation of $T_t\mathbb{R}$: acting on a germ $[f]$, it returns $\frac{d(f \circ \gamma)}{dt}(t)$. In a chart $(U, \varphi)$, the coordinates of the velocity are $\frac{d(\varphi \circ \gamma)_j}{dt}(t)$ by Theorem 3.2 (with the one-variable Jacobian a column of ordinary derivatives), which is exactly the derivative of the coordinate description of the curve: the abstract definition returns the familiar one, chart by chart. $\blacksquare$

Quick check.

1. Define the differential df_p.
2. Why is the differential independent of the representative g?
3. What is the matrix of the differential in chart bases?
4. State the chain rule on manifolds.
5. What is the velocity of a curve gamma at t?

Answers: 1. Applied to a germ g at the target, it evaluates X on the germ of g composed with f. 2. Representatives agreeing near the target compose with f to germs agreeing near p (continuity of f). 3. The Jacobian matrix of the chart representation (Theorem 3.2). 4. The differential of the composition is the composition of the differentials (Theorem 3.3). 5. The differential d gamma_t applied to the standard basis derivation.

Theorem 4.1 (local diffeomorphisms). Let $f: M \to N$ be a smooth map of $n$-dimensional manifolds, and let $df_p: T_pM \to T_{f(p)}N$ be a linear isomorphism for some $p$. Then $f$ restricts to a diffeomorphism from an open neighbourhood of $p$ onto an open neighbourhood of $f(p)$.

Proof. Choose charts $(U_1, \varphi)$ at $p$ and $(V, \psi)$ at $f(p)$, and shrink $U_1$ (Proposition 5.1 of 3A makes $f$ continuous, so $f^{-1}(V)$ is an open neighbourhood of $p$) so that $U := U_1 \cap f^{-1}(V)$ satisfies $f(U) \subseteq V$. The chart representation $F = \psi \circ f \circ \varphi^{-1}: \varphi(U) \to \psi(V)$ is smooth between open subsets of $\mathbb{R}^n$. By Theorem 3.2, the Jacobian matrix of $F$ at $\varphi(p)$ is the matrix of $df_p$ in the chart bases, which is invertible (an isomorphism between $n$-dimensional spaces has an invertible matrix in any bases; the bases here are the chart transports of Theorem 2.1). The Euclidean inverse function theorem (declared in 3A’s Scope) applies: there are open $A \ni \varphi(p)$ and $B \ni F(\varphi(p))$ with $F|_A: A \to B$ a diffeomorphism. Shrink once more: $U' := U \cap \varphi^{-1}(A)$ is an open neighbourhood of $p$, and

is a composition of smooth maps with smooth inverses (the outer maps are chart maps, smooth with smooth inverses on their images; the middle is the Euclidean diffeomorphism), hence by 3A’s composition propositions a diffeomorphism from the open $U'$ onto $f(U') = \psi^{-1}\big(F(A \cap \varphi(U))\big)$, which is open: $A \cap \varphi(U)$ is open in $A$, and $F|_A$ is a homeomorphism of $A$ onto the open set $B$, so the image is open in $B$ and hence in $\mathbb{R}^n$, containing $f(p)$. $\blacksquare$

Example (what the tangent space of the model space is). For $M = \mathbb{R}^n$ with its standard structure and the identity chart, Theorem 2.1’s transport is $\Sigma_v \mapsto v$ and the differential of a smooth $f: \mathbb{R}^n \to \mathbb{R}^n$ at $p$, read through the transports, is exactly the Euclidean derivative $f'(p)$: Theorem 3.2’s Jacobian matrix is the matrix of $f'(p)$ by 3A’s Proposition 1.1. The manifold definitions recover the Euclidean ones verbatim, which is the consistency check any new definition must pass. $\blacksquare$

Quick check.

1. State the inverse function theorem on manifolds.
2. Which two theorems does its proof assemble?
3. What does the differential of the identity do?
4. Why is the local diffeomorphism’s image open?
5. What is a diffeomorphism’s differential at each point?

Answers: 1. An isomorphism df_p makes f a local diffeomorphism at p (Theorem 4.1). 2. The Euclidean inverse function theorem; the chart transport isomorphism of Theorem 2.1. 3. It is the identity on the tangent space. 4. It is F applied to an open subset of its domain, and F is a homeomorphism there. 5. An isomorphism: the chain rule on f composed with its inverse gives the identity matrices.

1. Prove that a derivation $D$ at $a$ satisfies $D([f]) = 0$ for every germ $[f]$ with $[f](a) = 0$ and vanishing first partials at $a$. (§1 – write $f = f(a) + \sum (x_i - a_i) g_i$ by Hadamard and watch every term die.)
2. Compute $T_0\mathbb{R}$ explicitly: write the basis derivation $\frac{d}{dt}\big|_0$ acting on an arbitrary germ, and verify the transport of Theorem 2.1 is the identity. (§1, §2 – the chart is the identity.)
3. Prove that $df_p(X)$ is independent of the representative $g$ of the target germ. (§3 – if $g$ and $g'$ agree near $f(p)$, then $g \circ f$ and $g' \circ f$ agree near $p$, because $f$ is continuous at $p$ by 3A’s Proposition 5.1.)
4. Verify the change-of-basis law (Theorem 2.2) for the two stereographic charts of the circle $S^1$, at a point where both charts apply. (§2, 3A §4 – the transition is $t \mapsto 1/t$, with derivative $-1/t^2$; the law says the basis derivations transform by exactly that Jacobian entry.)
5. Prove that the differential of a constant map is the zero map. (§3 – $[g \circ c] = [g(c_0)]$ is a constant germ, and derivations kill constant germs by Lemma 1.1.)
6. Prove that a local diffeomorphism is an open map. (§4 – each point of the domain has a neighbourhood mapped diffeomorphically onto an open set; images of those neighbourhoods cover the image and are open.)
7. Prove that $d(\operatorname{id}_M)_p = \operatorname{id}_{T_pM}$. (§3 – act on $X$ and a germ $[g]$: $X([g]) = X([g])$.)

Problems. The quick checks above tested recall. These chain several results and require full written arguments; hints are included where the jump is large.

8. Prove that a diffeomorphism’s differential is an isomorphism at every point. (The chain rule applied to $f \circ f^{-1} = \operatorname{id}$ and $f^{-1} \circ f = \operatorname{id}$.)
9. Compute the velocity of $\gamma(t) = (\cos t, \sin t)$ on $S^1$ at $t$, in the stereographic chart $\varphi_S$. (The chart representation of $\gamma$ is $t \mapsto \cos t / (1 + \sin t)$; its ordinary derivative is the chart coordinate of the velocity by Theorem 3.2, giving $-1/(1 + \sin t)$.)

The entire page follows from one identification: differentiating is an algebraic operation. Once derivations are recognized as the algebraic shadow of directional differentiation (Theorem 1.3, powered by Hadamard’s lemma), they can be defined on any smooth manifold where no ambient arrow exists, and every classical object (basis vectors, the Jacobian, the chain rule, the inverse function theorem) re-emerges as a theorem about the transport of that shadow along charts.

• Level 3C: Differential Forms – forms as determinant evaluators, the exterior derivative, and the coordinate independence that makes them intrinsic, built on the differential of this page.
• Level 3A: Smooth Manifolds – the charts, smooth maps, and the Euclidean inverse function theorem used throughout this page.
• An Introduction to Manifolds – Loring Tu’s Chapter 3, whose derivation-based treatment of the tangent space this page follows, with the germ bookkeeping made explicit.