Defining the Differential
In elementary calculus the gradient of a smooth function \(f\) on \(\mathbb{R}^n\) is the vector
field whose components are the partial derivatives of \(f\),
\[
\operatorname{grad} f = \sum_{i=1}^{n} \frac{\partial f}{\partial x^i}\, \frac{\partial}{\partial x^i} .
\]
This expression does not survive a change of coordinates: it places the partial derivatives, which
are lower-indexed quantities, into the upper-indexed slots of a vector. Written out with an explicit
summation sign, the formula sidesteps the index convention rather than obeying it — the index
\(i\) appears downstairs on both factors, so no legitimate implicit summation could produce it
— and this is the visible symptom that the gradient, written this way, depends on the
coordinate system. The partial derivatives of a smooth
function are not the components of a coordinate-independent vector field. They are instead the
components of a
covector field,
and recognizing this is what makes the construction below coordinate-free.
The resolution is to let the derivative of \(f\) act directly on tangent vectors, producing a number
at each point, with no reference to a basis. A tangent vector at \(p\) is a derivation of smooth
functions; applying it to \(f\) and recording the result is a linear operation on tangent vectors,
hence a covector.
Definition: The Differential of a Function
Let \(M\) be a smooth manifold with or without boundary and let
\(f \in C^\infty(M)\) be a
smooth real-valued function.
The differential of \(f\) is the covector field \(df\) whose value at each
\(p \in M\) is the covector \(df_p \in T^*_pM\) defined by
\[
df_p(v) = vf, \qquad v \in T_pM .
\]
For fixed \(p\), the assignment \(v \mapsto vf\) is linear in \(v\), because tangent vectors act
linearly on functions; thus \(df_p\) is genuinely an element of the
cotangent space
\(T^*_pM\). What remains is to confirm that, as \(p\) varies, these covectors fit together into a
smooth section of the cotangent bundle.
Proposition: The Differential Is a Smooth Covector Field
For every \(f \in C^\infty(M)\), the differential \(df\) is a smooth covector field on \(M\).
Proof:
We apply the
smoothness criterion for covector fields
in the form requiring that \(df(X)\) be a smooth function for every smooth vector field \(X\).
For a
smooth vector field
\(X\) on \(M\), the function \(df(X)\) is given pointwise by
\[
df(X)(p) = df_p(X_p) = X_p f = (Xf)(p) ,
\]
so \(df(X) = Xf\). The action of a smooth vector field on a smooth function is again smooth, so
\(df(X)\) is smooth for every such \(X\). By the criterion, \(df\) is a smooth covector field.
The Coordinate Expression
To see what \(df\) is in coordinates, let \((x^i)\) be smooth coordinates on an open subset
\(U \subseteq M\), with corresponding coordinate
coframe
\((\lambda^i)\) on \(U\) — the dual basis fields written provisionally as \(\lambda^i\) when
the cotangent bundle was constructed. Writing \(df = A_i\,\lambda^i\) for component functions
\(A_i : U \to \mathbb{R}\), the defining identity \(A_i(p) = df_p\big(\partial/\partial x^i|_p\big)\)
evaluates to
\[
A_i(p) = df_p\!\left( \frac{\partial}{\partial x^i}\bigg|_p \right)
= \frac{\partial}{\partial x^i}\bigg|_p f
= \frac{\partial f}{\partial x^i}(p) .
\]
The component functions of \(df\) in any smooth chart are therefore the partial derivatives of \(f\)
with respect to those coordinates:
\[
df_p = \frac{\partial f}{\partial x^i}(p)\, \lambda^i|_p .
\]
This is the precise sense in which \(df\) is the coordinate-free counterpart of the classical
gradient: it packages the same partial derivatives, but as the lower-indexed components of a covector
field, where the index convention is satisfied and no dependence on coordinates is hidden.
Resolving the placeholder: the differentials of coordinate functions
Each coordinate function \(x^j : U \to \mathbb{R}\) is itself a smooth real-valued function, so it has
a differential \(dx^j\), a covector field on \(U\). Applying the coordinate formula just obtained to
\(f = x^j\) and using \(\partial x^j / \partial x^i = \delta^j_i\),
\[
dx^j\big|_p = \frac{\partial x^j}{\partial x^i}(p)\, \lambda^i|_p = \delta^j_i\, \lambda^i|_p
= \lambda^j|_p .
\]
The differential of the coordinate function \(x^j\) is exactly the coordinate basis covector
\(\lambda^j\). This identifies the placeholder introduced earlier with an intrinsic object, and from
here onward the coordinate coframe is written \((dx^i)\) rather than \((\lambda^i)\). The coordinate
expression for the differential takes its familiar form,
\[
df = \frac{\partial f}{\partial x^i}\, dx^i .
\]
In the one-dimensional case this collapses to the
elementary differential,
\[
df = \frac{df}{dx}\, dx ,
\]
now read not as an informal relation between infinitesimals, nor merely as the linear map on
displacements that the elementary definition provides, but as an identity between covector
fields on a one-dimensional manifold: \(dx\) is the basis covector field dual to \(d/dx\), and the
coefficient is the ordinary derivative.
A worked differential
For \(f(x, y) = x^2 y \cos x\) on \(\mathbb{R}^2\), the coordinate formula gives the differential
directly from the two partial derivatives,
\[
\begin{align*}
df &= \frac{\partial}{\partial x}\big( x^2 y \cos x \big)\, dx
+ \frac{\partial}{\partial y}\big( x^2 y \cos x \big)\, dy \\
&= \big( 2xy \cos x - x^2 y \sin x \big)\, dx + x^2 \cos x \, dy .
\end{align*}
\]
The differential is a single covector field on \(\mathbb{R}^2\); its two component functions are
the partial derivatives, attached to the coordinate covector fields \(dx\) and \(dy\). Evaluating
\(df_p\) on a tangent vector \(v = v^1\,\partial/\partial x + v^2\,\partial/\partial y\) returns
the directional derivative of \(f\) along \(v\), recovered as the component pairing
\(\partial_x f \cdot v^1 + \partial_y f \cdot v^2\).
Properties of the Differential
The differential inherits, at the level of covector fields, the algebraic rules obeyed by ordinary
derivatives. Each follows directly from the definition \(df_p(v) = vf\) together with the fact that
tangent vectors are derivations: linear and satisfying the product rule on smooth functions.
Proposition: Properties of the Differential
Let \(M\) be a smooth manifold with or without boundary, and let \(f, g \in C^\infty(M)\).
(a) Linearity. For constants \(a, b \in \mathbb{R}\),
\(d(af + bg) = a\,df + b\,dg\).
(b) Product rule. \(d(fg) = f\,dg + g\,df\).
(c) Quotient rule. On the open set where \(g \neq 0\),
\[
d\!\left( \frac{f}{g} \right) = \frac{g\,df - f\,dg}{g^2} .
\]
(d) Chain rule. If \(J \subseteq \mathbb{R}\) is an interval containing the
image of \(f\) and \(h : J \to \mathbb{R}\) is smooth, then
\(d(h \circ f) = (h' \circ f)\, df\).
(e) Constancy. If \(f\) is constant, then \(df = 0\).
Proof:
Each identity is verified by evaluating both sides on an arbitrary tangent vector \(v \in T_pM\)
at an arbitrary point and using that \(v\) is a derivation. For (a),
\[
d(af + bg)_p(v) = v(af + bg) = a\,(vf) + b\,(vg) = a\,df_p(v) + b\,dg_p(v) ,
\]
by linearity of \(v\). For (b), the derivation product rule gives
\[
d(fg)_p(v) = v(fg) = f(p)\,(vg) + g(p)\,(vf)
= \big( f\,dg + g\,df \big)_p(v) ,
\]
where the values \(f(p), g(p)\) are exactly the coefficients appearing in the covector fields
\(f\,dg\) and \(g\,df\) at \(p\). For (c), on the set where \(g \neq 0\) the
function \(f/g\) is smooth, and applying \(v\) to it through the derivation quotient rule yields
\(v(f/g) = \big( g(p)\,vf - f(p)\,vg \big)/g(p)^2\), which is the value of the asserted covector
field on \(v\).
For (d), it suffices to compare component functions in an arbitrary chart. By
the coordinate expression for the differential and the one-variable chain rule of ordinary
calculus,
\[
\frac{\partial (h \circ f)}{\partial x^i} = h'(f)\, \frac{\partial f}{\partial x^i} ,
\]
so \(d(h \circ f) = h'(f)\,\dfrac{\partial f}{\partial x^i}\, dx^i = (h' \circ f)\, df\). For
(e), if \(f\) is constant then \(vf = 0\) for every derivation \(v\), so
\(df_p(v) = 0\) for all \(v\) and all \(p\), giving \(df = 0\).
Part (d) is the differential's form of the chain rule, and it is the covector shadow of the chain
rule for the
differential of a smooth map,
\(d(h \circ f)_p = dh_{f(p)} \circ df_p\). Under the identification \(T_a\mathbb{R} \cong \mathbb{R}\)
reconciled at the end of this page, the one-variable map \(dh_{f(p)}\) is multiplication by the
number \(h'(f(p))\), so composing with it scales \(df_p\) by \(h' \circ f\); this is exactly the
factor appearing in (d). The converse of (e) requires an
additional hypothesis and is recorded separately, since the vanishing of a differential constrains a
function only up to the connectivity of its domain.
Proposition: Functions with Vanishing Differential
Let \(f\) be a smooth real-valued function on a smooth manifold \(M\) with or without boundary.
Then \(df = 0\) if and only if \(f\) is constant on each connected component of \(M\).
Proof:
It suffices to treat the case where \(M\) is connected and show that \(df = 0\) forces \(f\) to be
constant; the constant direction is part (e) above applied componentwise. Assume \(M\) is
connected and \(df = 0\). Fix \(p \in M\) and set
\(\mathcal{C} = \{ q \in M : f(q) = f(p) \}\). This set is nonempty and closed, the latter by
continuity of \(f\). It is also open: given any \(q \in \mathcal{C}\), choose a smooth coordinate
ball (or half-ball, when \(q \in \partial M\)) \(U\) about \(q\). On \(U\) the coordinate
expression of \(df = 0\) gives \(\partial f / \partial x^i \equiv 0\) for every \(i\), so by
elementary calculus \(f\) is constant on the connected set \(U\), hence equal to \(f(q) = f(p)\)
throughout \(U\); thus \(U \subseteq \mathcal{C}\). A nonempty subset of the connected space
\(M\) that is both open and closed is all of \(M\), so \(f \equiv f(p)\).
The Differential Along a Curve
The differential also packages the rate of change of a function along a moving point. If a smooth
curve traces a path through \(M\), feeding its velocity to \(df\) at each instant recovers the
ordinary derivative of the function restricted to the curve. This is the manifold form of the
statement that the directional derivative is the gradient paired with the velocity, freed of any
coordinate dependence.
Proposition: Derivative of a Function Along a Curve
Let \(M\) be a smooth manifold with or without boundary, let \(\gamma : J \to M\) be a smooth
curve on an interval \(J \subseteq \mathbb{R}\), and let \(f \in C^\infty(M)\). Then the
derivative of the real-valued function \(f \circ \gamma : J \to \mathbb{R}\) is
\[
(f \circ \gamma)'(t) = df_{\gamma(t)}\big( \gamma'(t) \big), \qquad t \in J ,
\]
where \(\gamma'(t)\) is the
velocity
of \(\gamma\) at \(t\).
Proof:
By the definition of the differential, \(df_{\gamma(t)}\) acts on the tangent vector
\(\gamma'(t) \in T_{\gamma(t)}M\) by applying that vector to \(f\):
\[
df_{\gamma(t)}\big( \gamma'(t) \big) = \gamma'(t)\, f .
\]
The velocity of a curve acts on a smooth function by differentiating the function along the
curve, that is, \(\gamma'(t)\,f = (f \circ \gamma)'(t)\). Combining the two equalities gives the
claim.
The differential as best linear approximation
The same object furnishes the linear approximation of \(f\) near a point. Working in coordinates on
an open subset \(U \subseteq \mathbb{R}^n\), regard \(f\) as a function of the coordinates and let
\(\Delta f = f(p + v) - f(p)\) be its increment along a displacement \(v\), the latter identified with
a tangent vector at \(p\) in the usual way. The first-order Taylor expansion gives
\[
\Delta f \approx \frac{\partial f}{\partial x^i}(p)\, v^i
= \frac{\partial f}{\partial x^i}(p)\, dx^i\big|_p(v)
= df_p(v) .
\]
Thus \(df_p\) is the linear functional that best approximates the increment of \(f\) near \(p\). The
strength of the construction is that this approximation is defined invariantly on any manifold,
without recourse to informal manipulation of infinitesimals: \(df_p\) is a genuine covector, and the
approximation statement is a statement about that covector.
Reconciliation with the differential of a smooth map
A smooth real-valued function \(f : M \to \mathbb{R}\) now carries two objects both written
\(df_p\). On one reading, \(f\) is a smooth map between manifolds, and its
differential as a smooth map
is the linear map \(df_p : T_pM \to T_{f(p)}\mathbb{R}\) into the tangent space of \(\mathbb{R}\) at
\(f(p)\). On the other, \(f\) is a real-valued function and its differential is the covector
\(df_p : T_pM \to \mathbb{R}\) of this chapter. These are the same object once the canonical
identification \(T_a\mathbb{R} \cong \mathbb{R}\) is taken into account, under which a tangent vector
\(c\,(d/dt|_a)\) corresponds to the real number \(c\). Both versions are represented in coordinates by
the same row matrix of partial derivatives of \(f\), which is why the single notation \(df_p\) serves
for both with no ambiguity in practice.
A parallel reconciliation holds for the expression \((f \circ \gamma)'(t)\) appearing in the
proposition. Viewing \(f \circ \gamma\) as a curve in \(\mathbb{R}\), its velocity is an element of
a tangent space \(T_{f(\gamma(t))}\mathbb{R}\); viewing it as a real-valued function of one variable,
its derivative is a real number. The two agree under the same identification
\(T_a\mathbb{R} \cong \mathbb{R}\), and the proposition expresses each equally well. Which reading is
in force is a matter of convenience, never of substance.
The critical points of a function
The general notion of a critical point of a smooth map — a point where the
differential fails to be surjective — specializes sharply for a real-valued function. Since
a linear map \(df_p : T_pM \to \mathbb{R}\) is surjective unless it is the zero map, \(p\) is a
critical point of \(f \in C^\infty(M)\) precisely when \(df_p = 0\). The vanishing of the covector
\(df_p\) is the coordinate-free statement that all first partial derivatives of \(f\) vanish at
\(p\), as the coordinate expression \(df_p = (\partial f / \partial x^i)(p)\, dx^i|_p\) makes
immediate: the covector is zero exactly when every component is. The differential thereby provides
the intrinsic notion of a stationary point on a manifold, independent of any chart, which is the
starting point for locating extrema and, more broadly, for the study of how the level sets of
\(f\) organize the manifold.