What Is a Vector Field?
Up to this point we have studied tangent vectors only at individual points. To describe motion, flow, or
any process that varies over a manifold — a velocity field on a fluid, an angular momentum field
on a rotating body, an infinitesimal symmetry of a Lie group — we need a tangent vector at
every point of the manifold, varying smoothly. This is the notion of a vector
field, the principal object of this chapter. Vector fields admit two complementary
descriptions, one geometric (arrows attached to points) and one algebraic (differential operators acting
on smooth functions), and the existence of globally nonvanishing vector fields turns out to be a subtle
topological question.
The definition
Recall that the
tangent bundle
\(TM\) of a smooth manifold \(M\) carries a natural projection \(\pi : TM \to M\) sending each tangent
vector to its base point, and that the
tangent bundle is itself a smooth manifold
of dimension \(2n\) when \(M\) has dimension \(n\). A vector field is a continuous right inverse to this
projection.
Definition: Vector Field
Let \(M\) be a smooth manifold with or without boundary. A vector field on \(M\) is
a continuous map \(X : M \to TM\), usually written \(p \mapsto X_p\), satisfying
\[
\pi \circ X = \mathrm{Id}_M ,
\]
or equivalently \(X_p \in T_pM\) for each \(p \in M\).
We write \(X_p\) rather than \(X(p)\) to remain consistent with the notation for elements of the tangent
bundle and to avoid collision with the action \(v(f)\) of a tangent vector on a function. Geometrically,
a vector field is what one pictures in elementary multivariable calculus: an arrow attached to each
point of the manifold, chosen tangent to \(M\) and varying continuously from point to point.
For most purposes the continuous category is too loose; the central object is the smooth one.
Definition: Smooth Vector Field
A vector field \(X : M \to TM\) is a smooth vector field if it is smooth as a map
between the smooth manifolds \(M\) and \(TM\).
It is occasionally useful to drop even continuity, retaining only the section property. The arguments
that follow will frequently start from such a candidate and conclude that it is in fact smooth.
Definition: Rough Vector Field
A rough vector field on \(M\) is a (not necessarily continuous) map
\(X : M \to TM\) with \(\pi \circ X = \mathrm{Id}_M\). Equivalently, a rough vector field assigns to
each \(p \in M\) some tangent vector \(X_p \in T_pM\), with no continuity assumed.
Finally, the locus where a vector field is nonzero is sometimes important to control.
Definition: Support of a Vector Field
The support of a vector field \(X\) on \(M\) is the closure of the set on which it
does not vanish,
\[
\mathrm{supp}\, X = \overline{\{ p \in M : X_p \neq 0 \}} .
\]
A vector field is compactly supported if its support is a compact subset of \(M\).
Coordinate expression and component functions
Let \((U, (x^i))\) be a smooth chart for \(M\). Because the
coordinate vectors
\(\partial/\partial x^i|_p\) form a basis for \(T_pM\) at every \(p \in U\), the value of any rough
vector field \(X\) at \(p \in U\) can be expanded uniquely as
\[
X_p = X^i(p) \left. \frac{\partial}{\partial x^i} \right|_p .
\]
The Einstein summation convention is in force throughout: a repeated index, once up and once down, is
summed from \(1\) to \(n = \dim M\). The \(n\) real-valued functions \(X^i : U \to \mathbb{R}\) thus
obtained are called the component functions of \(X\) in the chart \((U, (x^i))\). When
no confusion is possible we suppress the base point and write the identity at the level of vector
fields,
\[
X = X^i \frac{\partial}{\partial x^i} \qquad \text{on } U.
\]
The natural coordinates on \(TM\) constructed alongside its smooth structure record the base-point
coordinates of a tangent vector together with its components in the coordinate basis; in those
coordinates a vector field becomes a map into Euclidean space whose smoothness can be tested
componentwise.
Proposition (Smoothness Criterion for Vector Fields)
Let \(M\) be a smooth \(n\)-manifold with or without boundary, and let \(X : M \to TM\) be a rough
vector field. If \((U, (x^i))\) is any smooth chart on \(M\), the restriction \(X|_U\) is smooth if
and only if its component functions \(X^1, \dots, X^n\) with respect to that chart are smooth.
Proof:
Let \((x^i, v^i)\) denote the natural coordinates on \(\pi^{-1}(U) \subseteq TM\) associated with
the chart \((U, (x^i))\), under which a tangent vector \(v^i \partial/\partial x^i|_q\) at a point
\(q \in U\) corresponds to the tuple \((x^1(q), \dots, x^n(q), v^1, \dots, v^n)\). The restriction
\(X|_U\) takes values in \(\pi^{-1}(U)\), and in these coordinates its representation is
\[
\widehat{X}(x) = \bigl(x^1, \dots, x^n,\, X^1(x), \dots, X^n(x)\bigr) ,
\]
where \(X^i\) denotes the \(i\)th component function of \(X\) read in the \(x^i\)-coordinates. The
base-point entries are the identity in coordinates, so the smoothness of \(\widehat{X}\) is
equivalent to the smoothness of the fiber entries \(X^1, \dots, X^n\). Since smoothness of a map
between manifolds is by definition the smoothness of its coordinate representation in compatible
charts, the criterion follows.
First examples
The next four examples are the prototypes one returns to throughout the subject: the coordinate frame,
a globally defined nonvanishing field on \(\mathbb{R}^n\), and the global angle fields on circles and
tori.
Example (Coordinate Vector Fields):
Let \((U, (x^i))\) be any smooth chart on \(M\). For each fixed index \(i\) the assignment
\[
p \;\longmapsto\; \left. \frac{\partial}{\partial x^i} \right|_p
\]
defines a vector field on \(U\), the \(i\)th coordinate vector field, denoted
\(\partial/\partial x^i\). Its \(j\)th component function in the same chart is the Kronecker delta
\(\delta^i_j\), a constant, and so by the smoothness criterion above it is smooth on \(U\).
Example (The Euler Vector Field):
The Euler vector field on \(\mathbb{R}^n\) is the smooth vector field
\[
V_x = x^1 \left. \frac{\partial}{\partial x^1} \right|_x + \cdots + x^n \left. \frac{\partial}{\partial x^n} \right|_x ,
\]
whose component functions are the standard coordinates themselves. It vanishes at the origin and
points radially outward everywhere else, with magnitude growing linearly with the distance to the
origin. The name reflects its role in Euler's homogeneous function theorem, where a function \(f\)
on \(\mathbb{R}^n\) is positively homogeneous of degree \(k\) precisely when \(Vf = kf\), the action
of vector fields on functions being the one we introduce in the next section.
Example (The Angle Vector Field on the Circle):
Let \(\theta\) be any angle coordinate on a proper open subset \(U \subseteq \mathbb{S}^1\), and let
\(d/d\theta\) denote the corresponding coordinate vector field on \(U\). Any other angle coordinate
\(\widetilde\theta\) on an overlapping open set differs from \(\theta\) by an additive constant, so
the
component transformation law
for coordinate vectors gives \(d/d\theta = d/d\widetilde\theta\) on the overlap. The local
coordinate fields therefore agree on overlaps and assemble into a single smooth vector field defined
on the
circle \(\mathbb{S}^1\)
as a whole, which we still denote by \(d/d\theta\) by abuse of notation — strictly speaking,
the circle does not admit a global angle coordinate, so \(d/d\theta\) is not literally a coordinate
vector field over all of \(\mathbb{S}^1\). It is, however, smooth and nowhere vanishing, and it
records the canonical infinitesimal rotation of \(\mathbb{S}^1\) viewed as a
Lie group.
Example (Angle Vector Fields on Tori):
On the \(n\)-dimensional
torus \(\mathbb{T}^n\),
choosing an angle function \(\theta^i\) on the \(i\)th circle factor for each \(i = 1, \dots, n\)
yields local coordinates \((\theta^1, \dots, \theta^n)\) on \(\mathbb{T}^n\). The same argument
applied factor by factor produces \(n\) globally defined smooth vector fields
\(\partial/\partial\theta^1, \dots, \partial/\partial\theta^n\) on \(\mathbb{T}^n\), each of them
nowhere vanishing, together pointing along the \(n\) independent circular directions of the torus.
Vector fields along subsets and extension
The same flexibility that allowed smooth real-valued functions to be defined first on an arbitrary
subset and then extended to a neighborhood is available for vector fields, in essentially the same
form. The construction parallels the one used for
smooth maps on subsets:
on an arbitrary subset \(A \subseteq M\), one cannot meaningfully ask about differentiability at points
where \(A\) has no interior, and the workaround is to demand a smooth extension to an ambient open set.
Definition: Vector Field Along a Subset
Let \(A \subseteq M\) be an arbitrary subset of a smooth manifold with or without boundary. A
vector field along \(A\) is a continuous map \(X : A \to TM\) satisfying
\(\pi \circ X = \mathrm{Id}_A\); equivalently, \(X_p \in T_pM\) for each \(p \in A\). Such an
\(X\) is a smooth vector field along \(A\) if for every \(p \in A\) there exist a
neighborhood \(V\) of \(p\) in \(M\) and a smooth vector field \(\widetilde X\) on \(V\) with
\(\widetilde X = X\) on \(V \cap A\).
The substantive content is that, for a closed subset, smoothness along \(A\) is enough to guarantee a
smooth global extension — with control over the support, so the extension can be made to vanish
outside any prescribed open neighborhood of \(A\). This is the vector-field analogue of the extension
lemma for smooth functions, and the proof rests on the same machinery.
Lemma (Extension Lemma for Vector Fields)
Let \(M\) be a smooth manifold with or without boundary, let \(A \subseteq M\) be a closed subset,
and suppose \(X\) is a smooth vector field along \(A\). For any open set \(U \subseteq M\)
containing \(A\), there exists a smooth vector field \(\widetilde X\) on \(M\) such that
\(\widetilde X|_A = X\) and \(\mathrm{supp}\, \widetilde X \subseteq U\).
Proof Sketch:
The construction is the same partition-of-unity gluing used for the
extension lemma for smooth functions,
with vector-valued data in place of scalar-valued data. By the definition of smoothness along \(A\),
for each \(p \in A\) there is an open neighborhood \(V_p \subseteq M\) of \(p\) and a smooth vector
field \(\widetilde X_p\) on \(V_p\) agreeing with \(X\) on \(V_p \cap A\). Shrinking \(V_p\) if
necessary, we may assume \(V_p \subseteq U\). The open sets \(\{V_p : p \in A\}\) together with
\(V_0 := M \setminus A\) form an open cover of \(M\), where on \(V_0\) we declare the candidate
extension to be the zero vector field.
Apply the
existence of partitions of unity
to obtain a smooth partition of unity \(\{\psi_p\}_{p \in A} \cup \{\psi_0\}\) subordinate to this
cover, with \(\mathrm{supp}\, \psi_p \subseteq V_p\) and \(\mathrm{supp}\, \psi_0 \subseteq V_0\),
and define
\[
\widetilde X = \sum_{p \in A} \psi_p\, \widetilde X_p ,
\]
where each summand is extended by zero outside \(V_p\). Because the partition is locally finite, the
sum reduces near each point of \(M\) to a finite sum of smooth vector fields with smooth coefficients,
and is therefore a smooth vector field on \(M\). At a point \(q \in A\), every term contributing to
the sum has \(\widetilde X_p|_q = X_q\) (since \(q \in V_p \cap A\) whenever \(\psi_p(q) \neq 0\)),
so the partition-of-unity relation \(\sum_p \psi_p(q) + \psi_0(q) = 1\) together with the fact that
\(q \notin V_0\), giving \(\psi_0(q) = 0\), yields \(\widetilde X(q) = X(q)\). Finally, since each
\(\mathrm{supp}\, \psi_p \subseteq V_p \subseteq U\) and the partition is locally finite, the
support of \(\widetilde X\) is contained in the closure of the union, which lies in \(U\).
The smooth-function counterpart of this lemma was already used to produce smooth
bump functions
cutting off neighborhoods cleanly; here the same idea propagates one dimension up, from scalar values to
tangent vectors. A useful special case is the extension of a single tangent vector to a global vector
field, obtained by taking \(A\) to consist of one point.
Proposition (Pointwise Extension of Tangent Vectors)
Let \(M\) be a smooth manifold with or without boundary. For any point \(p \in M\) and any tangent
vector \(v \in T_pM\), there exists a smooth vector field \(X\) on \(M\) with \(X_p = v\).
Proof:
Apply the extension lemma above with \(A = \{p\}\) and \(U = M\). The single-point assignment
\(p \mapsto v\) is a vector field along \(A\), and it is smooth in the sense just defined: in any
smooth coordinate neighborhood \((W, (x^i))\) of \(p\) with coordinate representation
\(v = v^i \partial/\partial x^i|_p\), the constant-coefficient vector field
\(\widetilde X = v^i \partial/\partial x^i\) on \(W\) extends \(p \mapsto v\) smoothly. The extension
lemma then produces a smooth global vector field on \(M\) restricting to \(v\) at \(p\).
The \(C^\infty(M)\)-module \(\mathfrak{X}(M)\)
The set of all smooth vector fields on \(M\) carries additional algebraic structure beyond the
pointwise vector-space operations: one can multiply a smooth vector field by a smooth real-valued
function, scaling the vector at each point by the value of the function there. This compounds the
underlying real vector space into a module over the ring of smooth real-valued functions.
Definition: The Space \(\mathfrak{X}(M)\)
Let \(M\) be a smooth manifold with or without boundary. The set of all smooth vector fields on
\(M\) is denoted \(\mathfrak{X}(M)\). It is equipped with the pointwise operations
\[
(aX + bY)_p = a X_p + b Y_p \qquad (a, b \in \mathbb{R};\; X, Y \in \mathfrak{X}(M)),
\]
whose zero element is the vector field assigning to each \(p \in M\) the zero tangent vector
\(0 \in T_pM\). For a smooth real-valued function \(f \in C^\infty(M)\) and a smooth vector field
\(X \in \mathfrak{X}(M)\), the pointwise product
\[
(fX)_p = f(p)\, X_p
\]
defines a map \(fX : M \to TM\).
Two facts must be checked: that the pointwise operations produce smooth vector fields, and that the
resulting structure satisfies the module axioms.
Proposition (\(\mathfrak{X}(M)\) Is a Module)
Let \(M\) be a smooth manifold with or without boundary.
(a) For any \(X, Y \in \mathfrak{X}(M)\) and any \(f, g \in C^\infty(M)\), the map \(fX + gY\) is a
smooth vector field on \(M\).
(b) Under the operations above, \(\mathfrak{X}(M)\) is a module over the commutative ring
\(C^\infty(M)\).
Proof:
(a) In any smooth chart \((U, (x^i))\) the component functions of \(fX + gY\) are
\(f X^i + g Y^i\), with \(X^i, Y^i\) smooth on \(U\) by the smoothness criterion above and \(f, g\)
smooth by assumption; their linear combination is smooth, so by the same criterion \(fX + gY\) is
smooth on \(U\). Since the conclusion holds in any chart and smoothness of a map between manifolds
is a local condition, \(fX + gY\) is smooth on \(M\).
(b) Both operations are defined pointwise from the corresponding operations on the tangent spaces
\(T_pM\), which are real vector spaces. The module axioms — associativity and distributivity
of scalar multiplication over functions, compatibility with the ring structure on \(C^\infty(M)\),
and the unit law \(1 \cdot X = X\) — all reduce at each point \(p\) to the corresponding
identities for scalar multiplication on \(T_pM\), and so hold automatically.
Using the module structure, the coordinate expansion at a point can be lifted to an equation between
vector fields. If \(X \in \mathfrak{X}(M)\) and \((U, (x^i))\) is a smooth chart, the relation
\[
X = X^i \frac{\partial}{\partial x^i} \qquad \text{on } U
\]
holds in \(\mathfrak{X}(U)\), with \(X^i \in C^\infty(U)\) and \(\partial/\partial x^i \in \mathfrak{X}(U)\)
the coordinate vector fields. This is the form in which coordinate expressions for vector fields will
be used from now on.
Frames and Parallelizability
Coordinate vector fields provide a basis for the tangent space at each point of a chart, but they are
not the only way to do so. In many settings — orthonormal frames adapted to a metric, frames
adapted to a group structure, frames adapted to a submanifold — one wants tuples of vector fields
whose values at each point form a basis of the tangent space, without those tuples necessarily arising
from any coordinate chart. We give these tuples a name, record when they exist locally, and contrast
their local abundance with the global obstructions that distinguish manifolds from products.
Linear independence, spanning, and frames
Definition: Frame
Let \(M\) be a smooth \(n\)-manifold with or without boundary, and let \(A \subseteq M\). An ordered
\(k\)-tuple \((X_1, \dots, X_k)\) of vector fields defined on \(A\) is said to be linearly
independent on \(A\) if the tangent vectors \(X_1|_p, \dots, X_k|_p\) are linearly
independent in \(T_pM\) for every \(p \in A\), and to span the tangent bundle on \(A\)
if they span \(T_pM\) for every \(p \in A\). A local frame for \(M\) is an ordered
\(n\)-tuple \((E_1, \dots, E_n)\) of vector fields defined on an open subset \(U \subseteq M\) that
is both linearly independent and spans the tangent bundle on \(U\); equivalently,
\((E_1|_p, \dots, E_n|_p)\) is a basis of \(T_pM\) for every \(p \in U\). The frame is a
global frame if \(U = M\), and is smooth if each \(E_i\) is a
smooth vector field.
Because the cardinality of a basis of \(T_pM\) is fixed at \(n = \dim M\), an ordered \(n\)-tuple of
vector fields is a local frame as soon as it satisfies either of the two conditions: linear independence
alone forces spanning at each point (\(n\) independent vectors in an \(n\)-dimensional space form a
basis), and spanning alone forces linear independence (\(n\) vectors spanning an \(n\)-dimensional
space form a basis). We will use the shorthand \((E_i)\) for a frame when the index range is
understood.
Examples of frames
The four manifolds in the next set of examples are exactly the ones for which we have already produced
nonvanishing smooth vector fields in this chapter.
Examples (Local and Global Frames):
(a) On \(\mathbb{R}^n\), the standard coordinate vector fields
\((\partial/\partial x^1, \dots, \partial/\partial x^n)\) form a smooth global frame.
(b) For any smooth chart \((U, (x^i))\) on a smooth manifold \(M\) (possibly with boundary), the
coordinate vector fields \((\partial/\partial x^1, \dots, \partial/\partial x^n)\) form a smooth
local frame on \(U\), the coordinate frame associated with the chart. Every point
of \(M\) lies in the domain of some such local frame.
(c) The angle vector field \(d/d\theta\) on the circle \(\mathbb{S}^1\), constructed earlier in
this chapter, is itself a smooth global frame for \(\mathbb{S}^1\) — a single nonvanishing
smooth vector field, since \(\dim \mathbb{S}^1 = 1\).
(d) On the \(n\)-torus \(\mathbb{T}^n\), the \(n\) angle vector fields
\((\partial/\partial\theta^1, \dots, \partial/\partial\theta^n)\) form a smooth global frame.
Completion of local frames
A standard linear-algebra fact about a finite-dimensional vector space is that a linearly independent
set can be extended to a basis. The next proposition is the smooth-vector-field analogue: locally,
independent smooth vector fields can always be completed to a smooth local frame, and a basis of
\(T_pM\) at a single point can be realized as the values at \(p\) of a smooth local frame on a
neighborhood. The proof rests on the fact that linear independence is an open condition: the value of
a determinant is a smooth function of its entries, and the nonvanishing of a smooth function persists
on a neighborhood.
Proposition (Completion of Local Frames)
Let \(M\) be a smooth \(n\)-manifold with or without boundary.
(a) Suppose \((X_1, \dots, X_k)\) is a linearly independent \(k\)-tuple of smooth vector fields on
an open subset \(U \subseteq M\), with \(1 \le k < n\). For each \(p \in U\) there exist smooth
vector fields \(X_{k+1}, \dots, X_n\) on a neighborhood \(V \subseteq U\) of \(p\) such that
\((X_1, \dots, X_n)\) is a smooth local frame for \(M\) on \(V\).
(b) Suppose \((v_1, \dots, v_k)\) is a linearly independent \(k\)-tuple of vectors in \(T_pM\) for
some \(p \in M\), with \(1 \le k \le n\). There exists a smooth local frame \((X_i)\) on a
neighborhood of \(p\) with \(X_i|_p = v_i\) for \(i = 1, \dots, k\).
(c) Suppose \(A \subseteq M\) is closed and \((X_1, \dots, X_n)\) is a linearly independent
\(n\)-tuple of smooth vector fields along \(A\). There exists a smooth local frame
\((\widetilde X_1, \dots, \widetilde X_n)\) defined on some neighborhood of \(A\) such that
\(\widetilde X_i|_A = X_i\) for \(i = 1, \dots, n\).
Proof Sketch:
(a) Fix \(p \in U\). Choose a smooth chart \((W, (y^j))\) on a neighborhood
\(W \subseteq U\) of \(p\), with coordinate frame
\((\partial/\partial y^1, \dots, \partial/\partial y^n)\) on \(W\). The tangent vectors
\(X_1|_p, \dots, X_k|_p\) are linearly independent in \(T_pM\), and the coordinate vectors
\(\partial/\partial y^1|_p, \dots, \partial/\partial y^n|_p\) span \(T_pM\); a linearly
independent set in a finite-dimensional vector space can always be extended to a basis by adjoining
vectors from any spanning set, so there exist indices \(j_{k+1}, \dots, j_n \in \{1, \dots, n\}\)
such that the \(n\)-tuple
\(\bigl(X_1|_p, \dots, X_k|_p, \partial/\partial y^{j_{k+1}}|_p, \dots, \partial/\partial y^{j_n}|_p\bigr)\)
is a basis of \(T_pM\). Set \(X_i = \partial/\partial y^{j_i}\) on \(W\) for \(i = k+1, \dots, n\);
these are smooth vector fields on \(W\) by the example above.
Let \(A(q)\) denote the \(n \times n\) matrix whose columns are the coordinate representations of
\(X_1|_q, \dots, X_n|_q\) in the basis \((\partial/\partial y^j|_q)\). The function
\(q \mapsto \det A(q)\) is a smooth real-valued function on \(W\), and it is nonzero at \(q = p\)
by construction. Smoothness of the determinant function and continuity therefore give an open
neighborhood \(V \subseteq W\) of \(p\) on which \(\det A\) is nowhere zero, and on which the
\(n\)-tuple \((X_1, \dots, X_n)\) is therefore linearly independent at every point and so is a
smooth local frame for \(M\).
(b) In a smooth chart \((W, (x^i))\) centered at \(p\), write each \(v_j\) in the
coordinate basis as \(v_j = c^i_j \,\partial/\partial x^i|_p\), and define the smooth vector fields
\(X_j = c^i_j \,\partial/\partial x^i\) on \(W\) (constant coefficients in these coordinates), so
that \(X_j|_p = v_j\). By construction \((X_1, \dots, X_k)\) is linearly independent at \(p\) and
consists of smooth vector fields; shrinking \(W\) by the open-condition argument used in (a)
secures linear independence on a neighborhood, and applying (a) completes the tuple to a smooth
local frame on a possibly smaller neighborhood.
(c) Each component vector field \(X_i\) of the given \(n\)-tuple is smooth along
the closed subset \(A\) by hypothesis, so the
extension lemma for vector fields
produces a smooth vector field \(\widetilde X_i\) on \(M\) with \(\widetilde X_i|_A = X_i\). At
every point \(p \in A\) the values \(\widetilde X_1|_p, \dots, \widetilde X_n|_p\) coincide with
\(X_1|_p, \dots, X_n|_p\) and are therefore linearly independent. The open-condition argument from
(a) — applied this time pointwise to each \(p \in A\) using a coordinate chart at \(p\)
— gives an open neighborhood \(V_p \subseteq M\) of \(p\) on which the determinant of the
coordinate matrix of \((\widetilde X_1, \dots, \widetilde X_n)\) is nowhere zero. The union
\(V = \bigcup_{p \in A} V_p\) is then an open neighborhood of \(A\) on which
\((\widetilde X_1, \dots, \widetilde X_n)\) is a smooth local frame restricting to the given
\(n\)-tuple on \(A\).
Part (a) carries the main content of the three: a linearly independent tuple of smooth vector fields
on an open set can always be completed to a local frame on a neighborhood of any chosen point. Part (b)
lifts a single basis at one point to a smooth local frame realizing that basis as initial values. Part
(c) generalizes the pointwise statement (b) to a closed set, using the extension lemma to upgrade data
given along \(A\) to a frame defined in an open neighborhood of \(A\).
Orthonormal frames on subsets of \(\mathbb{R}^n\)
On subsets of \(\mathbb{R}^n\) the Euclidean dot product is available, and one can ask for frames whose
components are mutually perpendicular unit vectors at each point. This refinement is often more useful
than an arbitrary frame for problems with metric content.
Definition: Orthonormal Frame
Let \(A \subseteq \mathbb{R}^n\). A \(k\)-tuple of vector fields \((E_1, \dots, E_k)\) defined on
\(A\) is orthonormal if for every \(p \in A\) the tangent vectors
\(E_1|_p, \dots, E_k|_p\) are orthonormal in \(T_p\mathbb{R}^n\) with respect to the Euclidean dot
product, where \(T_p\mathbb{R}^n\) is identified with \(\mathbb{R}^n\) in the standard way. A
(local or global) frame on a subset of \(\mathbb{R}^n\) consisting of orthonormal vector fields is
called an orthonormal frame.
Beyond the standard coordinate frame, which is orthonormal everywhere on \(\mathbb{R}^n\), the simplest
nontrivial example pairs the radial direction with the angular one.
Example (Polar Orthonormal Frame on \(\mathbb{R}^2 \setminus \{0\}\)):
On the open subset \(\mathbb{R}^2 \setminus \{0\}\), set \(r = \sqrt{x^2 + y^2}\) and define
\[
E_1 = \frac{x}{r}\, \frac{\partial}{\partial x} + \frac{y}{r}\, \frac{\partial}{\partial y} ,
\qquad
E_2 = -\frac{y}{r}\, \frac{\partial}{\partial x} + \frac{x}{r}\, \frac{\partial}{\partial y} .
\]
Each component function is the quotient of a smooth function by \(r\), which is smooth and nowhere
vanishing on \(\mathbb{R}^2 \setminus \{0\}\), so \(E_1\) and \(E_2\) are smooth vector fields. A
direct computation gives
\(E_1 \cdot E_1 = (x^2 + y^2)/r^2 = 1\),
\(E_2 \cdot E_2 = (y^2 + x^2)/r^2 = 1\),
and \(E_1 \cdot E_2 = (-xy + xy)/r^2 = 0\),
verifying that \((E_1, E_2)\) is a smooth orthonormal frame on \(\mathbb{R}^2 \setminus \{0\}\).
Geometrically, \(E_1\) is the unit vector field tangent to the radial lines emanating from the
origin, and \(E_2\) is the unit vector field tangent to the circles centered at the origin,
oriented counterclockwise.
The Gram–Schmidt construction
Any local frame on an open subset of \(\mathbb{R}^n\) can be promoted to an orthonormal one by applying
the Gram–Schmidt procedure pointwise, the only thing to check being that the smooth dependence on
the base point is preserved.
Lemma (Gram–Schmidt Algorithm for Frames)
Let \(U \subseteq \mathbb{R}^n\) be open, and suppose \((X_1, \dots, X_n)\) is a smooth local frame
for \(T\mathbb{R}^n\) over \(U\). There exists a smooth orthonormal frame \((E_1, \dots, E_n)\) on
\(U\) such that
\[
\mathrm{span}\bigl(E_1|_p, \dots, E_j|_p\bigr)
= \mathrm{span}\bigl(X_1|_p, \dots, X_j|_p\bigr)
\qquad \text{for every } j = 1, \dots, n \text{ and every } p \in U .
\]
Proof:
Apply the classical Gram–Schmidt procedure to the vectors \((X_j|_p)\) at each \(p \in U\),
producing inductively the \(n\)-tuple of vector fields
\[
E_j = \frac{X_j - \sum_{i=1}^{j-1} (X_j \cdot E_i)\, E_i}
{\bigl| X_j - \sum_{i=1}^{j-1} (X_j \cdot E_i)\, E_i \bigr|}
\qquad (j = 1, \dots, n).
\]
At every \(p \in U\) and every \(j\), the vector
\(X_j|_p\) lies outside \(\mathrm{span}(X_1|_p, \dots, X_{j-1}|_p)\) because
\((X_1|_p, \dots, X_n|_p)\) is a basis of \(T_p\mathbb{R}^n\); the inductive hypothesis on the
spans matches this with
\(\mathrm{span}(E_1|_p, \dots, E_{j-1}|_p) = \mathrm{span}(X_1|_p, \dots, X_{j-1}|_p)\), so
\(X_j|_p\) lies outside the span of \(E_1|_p, \dots, E_{j-1}|_p\) as well, and the numerator of
\(E_j\) is therefore nonzero at every \(p \in U\). Its norm is then a smooth nowhere-vanishing
real-valued function on \(U\), the denominator is smooth, and the quotient is smooth on \(U\).
At each point, the resulting \(n\)-tuple is the Gram–Schmidt orthonormalization of
\((X_j|_p)\), which preserves the spans \(\mathrm{span}(X_1|_p, \dots, X_j|_p)\) as required.
Parallelizable manifolds
Smooth local frames exist in the neighborhood of every point of any smooth manifold, since every chart
supplies one. Smooth global frames, by contrast, often do not exist. The distinction has a name.
Definition: Parallelizable Manifold
A smooth manifold with or without boundary is parallelizable if it admits a smooth
global frame.
Several manifolds we have already encountered are parallelizable: \(\mathbb{R}^n\) under its standard
coordinate frame, the circle \(\mathbb{S}^1\) under \(d/d\theta\), the \(n\)-torus \(\mathbb{T}^n\)
under its tuple of angle vector fields. The spheres \(\mathbb{S}^3\) and \(\mathbb{S}^7\) are also
parallelizable, though the global frames in those cases require a separate construction we do not give
here; \(\mathbb{S}^3\) inherits one from its Lie group structure as the group of unit quaternions, and
\(\mathbb{S}^7\) inherits one from the unit octonions. It is known that \(\mathbb{S}^1\),
\(\mathbb{S}^3\), and \(\mathbb{S}^7\) are the only spheres of positive dimension that admit a smooth
global frame; the proof rests on tools from algebraic topology beyond the scope of the present text.
The lowest-dimensional sphere not on this list, \(\mathbb{S}^2\), is nonparallelizable: it does not
admit even a single smooth nowhere-vanishing vector field, let alone a global frame. The proof belongs
to algebraic topology and will not be addressed in this text; in the language of physical intuition,
this is the impossibility of combing the hair on a sphere flat.
Looking forward, we will see at the end of this chapter that every Lie group is parallelizable, the
left-translation structure of the group supplying a global frame automatically. The list of
parallelizable spheres above is consistent with this: of the spheres of positive dimension, exactly
\(\mathbb{S}^1\) and \(\mathbb{S}^3\) admit Lie group structures (as the unit complex numbers and the
unit quaternions, respectively), and these two are among the parallelizable ones; \(\mathbb{S}^7\) is
parallelizable but is not a Lie group, so parallelizability is necessary but not sufficient for a
smooth manifold to carry a Lie group structure.
Parallelizability as a Global Product Structure
For a manifold \(M\) covered by a single smooth chart, the tangent bundle \(TM\) was already shown
to be a
product \(M \times \mathbb{R}^n\):
the single chart globalizes the local product structure of the tangent bundle. Parallelizability
is the same phenomenon for manifolds requiring more than one chart. A smooth global frame
\((E_1, \dots, E_n)\) on \(M\) provides, at each point \(p\), a basis of \(T_pM\) that varies
smoothly with \(p\); the resulting map
\[
M \times \mathbb{R}^n \longrightarrow TM, \qquad
(p, v^1, \dots, v^n) \longmapsto v^i E_i\big|_p ,
\]
is a smooth bijection, and a closer analysis — whose natural setting is the general theory
of vector bundles, taken up in a later chapter — confirms that it is a diffeomorphism.
Parallelizability is therefore equivalent to the tangent bundle being globally a product, and the
failure of most manifolds to be parallelizable is the same as the failure of their tangent
bundles to split globally as products: the recurring local-versus-global distinction that vector
bundle theory makes precise.
Vector Fields as Derivations
A vector field on a manifold has, so far, been treated as a geometric object: a smoothly varying arrow
at each point of \(M\). There is a second description, algebraic rather than geometric, in which a
smooth vector field is an operator on the space of smooth real-valued functions. The two descriptions
are equivalent — identifying them is the main statement of this section — and each is
needed at different stages of the theory. Coordinate computations and pictures are easier in the
geometric language; combinatorial identities and the Lie bracket are easier in the algebraic one.
The action of a vector field on a function
Given a smooth vector field \(X \in \mathfrak{X}(M)\) and a smooth real-valued function
\(f \in C^\infty(U)\) on an open subset \(U \subseteq M\), we obtain a new real-valued function
\(Xf : U \to \mathbb{R}\) by letting the tangent vector \(X_p\) act on \(f\) at each point:
\[
(Xf)(p) = X_p f .
\]
The right-hand side uses the standard action of a tangent vector on a smooth function defined in a
neighborhood of the point, that is, the action of a
derivation at a point.
The Notations \(fX\) and \(Xf\) Are Not the Same
Two different objects share the same two letters in different orders:
\(fX\) is a smooth vector field on \(U\), the scalar multiple of \(X\) by the
smooth function \(f\). It is an element of \(\mathfrak{X}(U)\), and at each point \(p\) its value
is the tangent vector \(f(p) X_p \in T_pM\).
\(Xf\) is a real-valued function on \(U\), the result of letting the vector field
\(X\) act on the function \(f\). It is an element of \(C^\infty(U)\) once we have shown it is
smooth, and at each point \(p\) its value is the real number \(X_p f \in \mathbb{R}\).
The first is a vector-valued object, the second is a scalar-valued one; they are interchanged
nowhere. The notational similarity cannot be avoided and is a source of recurring confusion,
especially in formulas mixing both, and we ask the reader to keep the distinction firmly in mind.
Because the action of a tangent vector on a smooth function depends only on the values of the function
in an arbitrarily small neighborhood of the point — the
locality property of tangent vectors
— the function \(Xf\) is locally determined as well. Concretely, if \(V \subseteq U\) is open and
\(f|_V\) denotes the restriction, then
\[
(Xf)|_V = X(f|_V) ,
\]
where the right-hand side is the action of \(X\) on the function \(f|_V \in C^\infty(V)\). The
statement is that restriction commutes with the action of the vector field. The next proposition uses
this locality to give an additional criterion for the smoothness of \(X\), couched in the algebraic
description.
The smoothness criterion via the action on functions
Proposition (Smoothness Criterion via the Action on Functions)
Let \(M\) be a smooth manifold with or without boundary, and let \(X : M \to TM\) be a rough vector
field. The following are equivalent:
(a) \(X\) is a smooth vector field.
(b) For every \(f \in C^\infty(M)\), the function \(Xf\) is smooth on \(M\).
(c) For every open set \(U \subseteq M\) and every \(f \in C^\infty(U)\), the function \(Xf\) is
smooth on \(U\).
Proof:
We prove the implications (a) \(\Rightarrow\) (b) \(\Rightarrow\) (c) \(\Rightarrow\) (a).
(a) \(\Rightarrow\) (b). Suppose \(X\) is smooth and \(f \in C^\infty(M)\). For
any point \(p \in M\), choose a smooth chart \((U, (x^i))\) on a neighborhood of \(p\). On \(U\)
the tangent vector \(X_q\) at \(q \in U\) acts on \(f\) by the
coordinate basis
expression
\[
X_q f
= \left( X^i(q) \left. \frac{\partial}{\partial x^i} \right|_q \right) f
= X^i(q)\, \frac{\partial f}{\partial x^i}(q) ,
\]
where \(X^i\) are the component functions of \(X\) in the chart. The component functions \(X^i\)
are smooth on \(U\) by the
smoothness criterion for vector fields,
and the partial derivatives \(\partial f/\partial x^i\) are smooth on \(U\) because \(f\) is. Their
sum \(Xf = X^i \,\partial f/\partial x^i\) is therefore smooth on \(U\). Since every point of \(M\)
has such a neighborhood and smoothness is a local condition, \(Xf\) is smooth on \(M\).
(b) \(\Rightarrow\) (c). Suppose every \(Xf\) with \(f \in C^\infty(M)\) is
smooth, and let \(U \subseteq M\) be open and \(f \in C^\infty(U)\). For each \(p \in U\), choose
a
smooth bump function
\(\psi \in C^\infty(M)\) equal to \(1\) on a neighborhood \(V_p \subseteq U\) of \(p\) and
compactly supported in \(U\). The product \(\widetilde f = \psi f\), extended by zero outside
\(\mathrm{supp}\, \psi\), is a smooth function on all of \(M\) and agrees with \(f\) on \(V_p\).
By hypothesis \(X\widetilde f\) is smooth on \(M\), and by the locality identity above,
\(Xf\) agrees with \(X\widetilde f\) on \(V_p\). Hence \(Xf\) is smooth on a neighborhood of every
point of \(U\), and so smooth on \(U\).
(c) \(\Rightarrow\) (a). Suppose \(Xf\) is smooth on every open set whenever the
function being acted on is smooth there, and let \((U, (x^i))\) be a smooth chart on \(M\). Each
coordinate function \(x^i\) is smooth on \(U\), so by hypothesis \(X x^i\) is smooth on \(U\).
Evaluating the coordinate-basis expression of \(X_q\) on \(f = x^i\) gives
\[
(X x^i)(q)
= X_q x^i
= X^j(q) \left. \frac{\partial x^i}{\partial x^j} \right|_q
= X^j(q)\, \delta^i_j
= X^i(q) ,
\]
showing that the component function \(X^i\) coincides with \(Xx^i\) and is therefore smooth on
\(U\). All component functions of \(X\) in any chart are smooth, so by the smoothness criterion
for vector fields, \(X\) is a smooth vector field on \(M\).
The product rule and derivations
The action of a tangent vector on a smooth function satisfies a product rule at each point: this is
one of the
defining properties of derivations
in the pointwise sense. Translating this pointwise statement into an identity between smooth functions
yields a product rule for the action of a vector field on a product of smooth functions.
For \(X \in \mathfrak{X}(M)\) and \(f, g \in C^\infty(M)\), the function \(X(fg) \in C^\infty(M)\)
satisfies, at every point \(p \in M\),
\[
\bigl( X(fg) \bigr)(p)
= X_p (fg)
= f(p)\, X_p g + g(p)\, X_p f
= f(p) (Xg)(p) + g(p) (Xf)(p) ,
\]
by the pointwise product rule. Since this holds at every \(p\), it is the equality of smooth functions
\[
X(fg) = f\, Xg + g\, Xf .
\]
This identity, together with \(\mathbb{R}\)-linearity, abstracts to the following notion.
Definition: Derivation of \(C^\infty(M)\)
Let \(M\) be a smooth manifold with or without boundary. A derivation of \(C^\infty(M)\)
is an \(\mathbb{R}\)-linear map \(D : C^\infty(M) \to C^\infty(M)\) satisfying the product rule
\[
D(fg) = f\, Dg + g\, Df \qquad \text{for all } f, g \in C^\infty(M) .
\]
The terminology coincides in name but not in type with the notion of derivation at a point, used
earlier as one of the characterizations of a tangent vector: a derivation at \(p\) is an
\(\mathbb{R}\)-linear map from \(C^\infty(M)\) into \(\mathbb{R}\) satisfying the product rule with the
values of \(f\) and \(g\) at \(p\) as coefficients, whereas a derivation in the present sense maps
\(C^\infty(M)\) into itself with the full functions \(f\) and \(g\) as coefficients. The two notions
are linked by the next result: every derivation of \(C^\infty(M)\) is the action of a uniquely
determined smooth vector field, recovered pointwise as a derivation at each point.
The identification of vector fields with derivations
Proposition (Smooth Vector Fields Are Derivations of \(C^\infty(M)\))
Let \(M\) be a smooth manifold with or without boundary. A map
\(D : C^\infty(M) \to C^\infty(M)\) is a derivation if and only if it is of the form \(Df = Xf\)
for a uniquely determined smooth vector field \(X \in \mathfrak{X}(M)\).
Proof:
Suppose first that \(X \in \mathfrak{X}(M)\) is a smooth vector field and \(Df := Xf\) for
\(f \in C^\infty(M)\). The smoothness criterion for the action on functions, established above,
gives \(Xf \in C^\infty(M)\) whenever \(f \in C^\infty(M)\), so \(D\) maps \(C^\infty(M)\) into
itself. \(\mathbb{R}\)-linearity of \(D\) follows from the \(\mathbb{R}\)-linearity of each
\(X_p : C^\infty(M) \to \mathbb{R}\), and the product rule \(D(fg) = f\, Dg + g\, Df\) is the
function-level identity derived above. Hence \(D\) is a derivation of \(C^\infty(M)\).
Conversely, suppose \(D : C^\infty(M) \to C^\infty(M)\) is a derivation. If \(D\) is to be of the
form \(Df = Xf\) for some vector field \(X\), the value at \(p\) of the tangent vector \(X_p\) is
forced: for every \(f \in C^\infty(M)\),
\[
X_p f = (Xf)(p) = (Df)(p) .
\]
We take this as the definition. For each \(p \in M\), define
\(X_p : C^\infty(M) \to \mathbb{R}\) by \(X_p f := (Df)(p)\). The map \(X_p\) is
\(\mathbb{R}\)-linear because evaluation at \(p\) is \(\mathbb{R}\)-linear and \(D\) is
\(\mathbb{R}\)-linear, and it satisfies the pointwise product rule because
\[
X_p(fg) = \bigl( D(fg) \bigr)(p) = \bigl( f\, Dg + g\, Df \bigr)(p) = f(p) X_p g + g(p) X_p f .
\]
So \(X_p\) is a derivation at \(p\), and hence an element of the
tangent space \(T_pM\),
which is defined to be the space of all such derivations. The assignment \(p \mapsto X_p\) is then
a (rough) vector field, and \(Xf = Df\) for every \(f \in C^\infty(M)\) by construction. Since
\(Df \in C^\infty(M)\) for every \(f \in C^\infty(M)\), criterion (b) of the smoothness criterion
via the action on functions is satisfied, and \(X\) is therefore a smooth vector field. Uniqueness
follows from the same forced formula: any vector field \(X'\) with \(X'f = Df\) for all \(f\) must
have \(X'_p f = (Df)(p) = X_p f\) for all \(p\) and \(f\), so \(X' = X\).
The upshot is that smooth vector fields on \(M\) are the same things as derivations of \(C^\infty(M)\),
and we will pass between the two descriptions freely. When this identification is being used, the same
letter \(X\) denotes both the geometric object (a smooth section of the tangent bundle, a map from \(M\)
into \(TM\)) and the algebraic operator (an \(\mathbb{R}\)-linear derivation from \(C^\infty(M)\) into
itself). The composition of two such operators, and the question of when that composition or its
antisymmetrization is again a derivation, is the elementary observation that opens onto the Lie bracket
and the further algebraic structure of \(\mathfrak{X}(M)\) developed in the next stages of this theory.
Velocity Fields, Flows, and Learned Dynamics
The vector-field formalism developed in this chapter has the unusual feature of being both a historical
origin of the subject and an actively used language outside mathematics. We close by indicating three
settings in which vector fields are the primary modeling object: the classical description of
continuous motion in fluids and rigid bodies, the configuration-space description of robotic systems,
and a recent class of generative models in machine learning whose sampling dynamics are governed by a
learned, time-dependent vector field. The three settings share a structural feature that this chapter
has made formal — an assignment of a tangent vector to each point of a state space, varying
smoothly with the point — but they exploit it in quite different ways.
Velocity fields in continuum mechanics
A classical setting for vector fields is continuum mechanics, where the instantaneous state of motion
of a fluid or deformable body is described by an Eulerian velocity field: a smooth assignment, to each
point of the region occupied by the medium, of the velocity of the particle currently at that point.
On a domain \(\Omega \subseteq \mathbb{R}^3\) this is a smooth vector field
\(v \in \mathfrak{X}(\Omega)\), and the governing equations of fluid mechanics — Euler's
equations for inviscid flow, the Navier–Stokes equations for viscous flow — are partial
differential equations for \(v\) and the associated pressure and density fields. The trajectories of
individual particles are the integral curves of \(v\), in the sense developed when ordinary
differential equations are studied on manifolds in the next stage of this theory.
The same formalism describes rigid-body kinematics on richer state spaces. The configuration of a rigid
body in three dimensions is a point of the group of rigid motions \(SE(3)\), a six-dimensional manifold;
the instantaneous motion is a tangent vector at the current configuration, and an evolving motion is an
integral curve of a vector field on \(SE(3)\). The shift from \(\mathbb{R}^3\) to \(SE(3)\) is forced
by the geometry — rotations do not form a Euclidean space — and the abstract machinery of
the present chapter is what makes the shift tractable.
Configuration-space vector fields in robotics and control
A robotic arm with \(n\) revolute joints has a configuration space \(\mathbb{T}^n\): a single
configuration is specified by \(n\) joint angles, each living on a circle. The angle vector fields on
the torus constructed earlier in this chapter are then literally interpretable: each
\(\partial/\partial\theta^i\) represents the infinitesimal motion that rotates the \(i\)th joint while
leaving the others fixed, and a generalized velocity of the arm at a given configuration is a tangent
vector at that point of \(\mathbb{T}^n\). For more complex mechanisms the configuration space is some
other smooth manifold — \(SE(3)\) for a free-floating rigid body, products of such groups for
serial chains, and more intricate manifolds for systems with closed kinematic loops — and the
same language of vector fields supplies the velocity description.
A control system in this formalism is typically specified by giving a finite list of vector fields on
the configuration manifold, together with a way of combining them according to control inputs; the
questions of which configurations are reachable from a given starting point and how to plan a motion
are then questions about the integral curves and combinations of these vector fields, drawing on the
Lie-bracket structure introduced in the next stage of this theory.
Symbolic computation and the operator description
The identification of smooth vector fields with derivations of \(C^\infty(M)\) is well suited to
symbolic implementation. In computer algebra systems, a vector field on an open subset of
\(\mathbb{R}^n\) is represented by its \(n\) component functions in a chart, and its action on a
function is implemented as the corresponding linear combination of partial-derivative operators, in
exact correspondence with the formula
\(Xf = X^i \,\partial f / \partial x^i\) established in this section. The operator picture also makes
composition meaningful: two vector fields, viewed as derivations, can be applied in succession, and
this composition is the starting point for the Lie bracket construction. Symbolic algebra packages for
differential geometry implement both the geometric and the algebraic pictures, switching between them
as the computation demands — the same equivalence whose proof occupied the previous part of this
chapter.
Flow-based generative models in machine learning
A recent family of generative models in machine learning — continuous-time normalizing flows,
neural ordinary differential equations, and flow matching among them — is built on vector fields
in a setting closer to classical dynamics than the earlier neural-network architectures. We describe
the structure abstractly, with the caveat that what such models accomplish empirically is a separate
matter from the vector-field machinery they use to do it.
Sampling as Integration of a Learned Vector Field
In this family of models, the state space is taken to be \(\mathbb{R}^d\), and a sample is produced
by transporting a point drawn from a fixed prior distribution (commonly an isotropic Gaussian) to
a point in the data distribution. The transport is governed by an ordinary differential equation
\[
\frac{d}{dt}\, h(t) = v_\theta\bigl( h(t),\, t \bigr) ,
\qquad h(0) \sim p_{\mathrm{prior}} ,\quad t \in [0, 1] ,
\]
where \(v_\theta : \mathbb{R}^d \times [0, 1] \to \mathbb{R}^d\) is a time-dependent vector field
on \(\mathbb{R}^d\) whose parameters \(\theta\) are learned from data. For each fixed \(t\) the map
\(h \mapsto v_\theta(h, t)\) is a vector field on \(\mathbb{R}^d\) in the sense of this chapter;
the time dependence places the joint object \(v_\theta\) in the broader class of time-dependent
vector fields, whose theory extends the autonomous case treated in subsequent stages of this
development. Sampling at inference time consists of numerically integrating this ODE from \(t = 0\)
to \(t = 1\). Training adjusts \(\theta\) so that the resulting transport pushes the prior onto
the data distribution; the training objective differs across formulations (likelihood-based for
continuous normalizing flows, regression against a tractable target field for flow matching), and
a separate body of theory addresses when and how well such training succeeds.
Two warnings are in order. First, the state space here is the Euclidean space \(\mathbb{R}^d\)
carrying no preferred geometric structure beyond that of a smooth manifold; the more elaborate
machinery of Lie groups and curved manifolds is not in use in the baseline formulations, though
extensions to nontrivial manifolds and group-equivariant variants are an active research area.
Second, this construction is a single family among many in modern generative modeling, alongside
autoregressive language models, variational autoencoders, and others that do not have a
vector-field formulation; the formalism of this chapter is the natural language for this family,
not a foundation for generative modeling in general.
These four threads — continuum mechanics, configuration-space dynamics, symbolic differential
operators, and learned flows — do not exhaust the uses of vector fields, but they illustrate the
breadth of the object introduced in this chapter. The remaining parts of this development extend the
formalism in two complementary directions: outward, to the way vector fields interact with smooth maps
between manifolds and with submanifolds, and inward, to the algebraic structure on \(\mathfrak{X}(M)\)
organized by the Lie bracket.