Def: A (local or global) section of $T^{\ast }M$ is called a covector field or differential $1$-form. Like sections of other bundles, covector fields without further qualifications are assumed to be merely continuous; when we make different assumptions, we use rough vector field and smooth vector field. In any smooth local coordinates on an open subset $U\subseteq M$, a rough covector field $\omega$ can be written in terms of the coordinate covector fields $(d{x}^i)$ as $\omega ={\omega }_{i}d{x}^i$ for $n$ functions ${\omega }_i:U\to \mathbb{R}$ called the component functions of $\omega$. They are characterised by $$\omega_i(p) := \omega_p\left(\left.\frac{\partial}{\partial x^i}\right\rvert_p\right).$$
If $\omega$ is a rough covector field and $X$ is a vector field on $M$, then we can form a function $\omega (X):M\to \mathbb{R}$ by $$\omega(X)(p) := \omega_p(X_p),\qquad p\in M. $$If we write $\omega ={\omega }_{i}d{x}^i$ and $X=X^j\partial /\partial x^j$ in terms of local coordinates, then $\omega (X)={\omega }_i{X}^i$.

Smoothness Criteria for Covector Fields: Let $M$ be a smooth manifold with or without boundary, and let $\omega :M\to T^{\ast }M$ be a rough covector field. The following are equivalent:

• $\omega$ is smooth.
• In every smooth coordinate chart, the component functions are smooth.
• Each point of $M$ is contained in some coordinate chart in which $\omega$ has smooth component functions.
• For every smooth vector field $X\in \mathfrak{X}(M)$, the function $\omega (X)$ is smooth on $M$.
• For every open subset $U\subseteq M$ and every vector field $X$ on $U$, the function $\omega (X):U\to \mathbb{R}$ is smooth on $U$.

Coframes

Def: Let $M$ be a smooth manifold with or without boundary, and let $U\subseteq M$ be an open subset. A local coframe for $M$ over $U$ is an ordered $n$-tiple of covector fields $({\epsilon }^1,\dots ,{\epsilon }^n)$ defined on $U$ such that $({\epsilon }^i{|}_p)$ forms a basis for $T_p^{\ast }M$ at each $p\in M$. If $U=M$, it is called a global coframe.

For any smooth chart $(U,(x^i))$, the coordinate covector fields $(d{x}^i)$ constitute a local coframe over $U$, called a coordinate coframe.

Def: Given a local frame $(E_1,\dots ,E_n)$ for $TM$ over an open subset $U$, there is a uniquely determined (rough) local coframe $({\epsilon }^1,\dots ,{\epsilon }^n)$ over $U$ such that $({\epsilon }^i{|}_p)$ is the dual basis to $(E_i{|}_p)$ for each $p\in U$, or equivalently ${\epsilon }^i(E_j)={\delta }_j^i$. This coframe is called the coframe dual of $(E_i)$. Conversely, if we start with the local coframe $({\epsilon }^i)$ over an open subset $U\subseteq M$, there is a uniquely determined local coframe $(E_i)$, called the frame dual to $({\epsilon }^i)$, determined by ${\epsilon }^i(E_j)={\delta }_j^i$.

Lemma: Let $M$ be a smooth manifold with or without boundary. If $(E_i)$ is a rough local frame over an open subset $U\subseteq M$ and $({\epsilon }^i)$ is its dual coframe, then $(E_i)$ is smooth iff $({\epsilon }^i)$ is smooth.

Given a local coframe $({\epsilon }^i)$ over an open subset $U\subseteq M$, every rough covector field $\omega$ on $U$ can be expressed in terms of the coframe as $\omega ={\omega }_i{\epsilon }^i$ for some functions ${\omega }_1,\dots ,{\omega }_n:U\to \mathbb{R}$, called the component functions of $\omega$ with respect to the given coframe. The component functions are determined by ${\omega }_i:=\omega (E_i)$ where $(E_i)$ is the frame dual to $({\epsilon }^i)$.

Coframe Criterion for Smoothness of Covector Fields: Let $M$ be a smooth manifold with or without boundary, and let $\omega$ be a rough covector field on $M$. If $({\epsilon }^i)$ is a smooth coframe on an open subset $U\subseteq M$, then $\omega$ is smooth on $U$ iff its component functions with respect to $({\epsilon }^i)$ are smooth.

We denote the ${\mathcal{C}}^{\mathrm{\infty }}$-module of all smooth covector fields on $M$ by ${\mathfrak{X}}^{\ast }(M)$.

Differential of a Function

Def: Let $f$ be a smooth real-valued function on a smooth manifold $M$ with or without boundary. We define a covector field $df$, called the differential of $f$, by $$df_p(v) := vf, \qquad v\in T_p M. $$
Prop: The differential of a smooth function is a smooth covector field.

Let $(x^i)$ be smooth coordinates on an open subset $U\subseteq M$, and let $({\lambda }^i)$ be the correspinding coordinate frame on $U$, the corresponding coframe of $(\partial /\partial x^i)$. If we write $d{f}_p=A_i(p){\lambda }^i{|}_p$ for some functions $A_i:U\to \mathbb{R}$; then $$A_i(p) = df_p\left(\left.\frac{\partial}{\partial x^i}\right\rvert_p\right) = \frac{\partial f}{\partial x^i}(p).$$This means that$$df_p = \frac{\partial f}{\partial x^i} (p)\lambda^i|_p.$$If we apply this identity to the the special case in which $f$ is one of the coordinate functions $x^j:U\to \mathbb{R}$, we obtain that$$dx^j|_p = \frac{\partial x^j}{\partial x^i}(p)\lambda^i|_p = \delta^j_i \lambda^i|_p = \lambda^j|_p.$$In other words, the coordinate covector field ${\lambda }^j$ is none other than the differential $d{x}^j$, which explains the notation so far used, and a nice formula for the differential of a function in terms of local coordinates $$df = \frac{\partial f}{\partial x^i} dx^i. $$
Prop: Suppose $M$ is a smooth $n$-manifold, $p\in M$, and $y^1,\dots ,y^k$ are smooth real valued functions defined on a neighbourhood of $p\in M$.

• If $k=n$ and $(d{y}^1{|}_p,\dots ,d{y}^n{|}_p)$ is a basis for $T_p^{\ast }M$, then $(y^1,\dots ,y^n)$ are smooth coordinates for $M$ in some neighbourhood of $p$.
• If $(d{y}^1{|}_p,\dots ,d{y}^k{|}_p)$ is a linearly independent $k$-tuple of covectors and $k<n$, then are some function $y^{k+1},\dots ,y^n$ such that $(y^1,\dots ,y^n)$ are smooth coordinates for $M$ in neighbourhoods of $p$.
• If $(d{y}^1{|}_p,\dots ,d{y}^k{|}_p)$ span $T_p^{\ast }M$, there are indices $i_1,\dots ,i_n$ such that $(y^{i_1},\dots ,y^{i_n})$ are smooth coordinates for $M$ in a neighbourhood of $p$.

Properties of the Differential: Let $M$ be a smooth manifold with or without boundary, and let $f,g\in {\mathcal{C}}^{\mathrm{\infty }}(M)$.

• if $a$ and $b$ are constants, then $d(af+bg)=adf+bdg$.
• $d(fg)=fdg+gdf$.
• $d(f/g)=(gdf-fdg)/g^2$ on the set where $g\ne 0$.
• If $J\subseteq \mathbb{R}$ is an interval containing the image of $f$, and $h:J\to \mathbb{R}$ is a smooth function, then $d(h\circ f)=(h^{\prime }\circ f)df$.
• if $f$ is constant, then $df=0$.

Functions with Vanishing Differential: If $f$ is a smooth real-valued function on a smooth manifold $M$ with or without boundary, then $df=0$ iff $f$ is constant on each components of $M$.

Derivative of a Function Along a Curve: Suppose $M$ is a smooth manifold with or without boundary, $\gamma :J\to M$ is a smooth curve, and $f:M\to \mathbb{R}$ is a smooth function. Then the derivative of the real-valued function $f\circ \gamma :J\to \mathbb{R}$ is given by $$(f\circ \gamma)'(t) := df_{\gamma(t)}(\gamma'(t)). $$

Prop: For a real-valued function $f:M\to \mathbb{R}$, we have that $p\in M$ is a critical point of $f$ iff $d{f}_p=0$.

There might be a little bit of a problem since the differential for a smooth real-valued function $f:M\to \mathbb{R}$ is a little ambiguous. We can think of it as the linear transformation from $d{f}_p:T_{p}M\to T_{f(p)}\mathbb{R}$ and the covector $d{f}_p:T_{p}M\to \mathbb{R}$, but in reality they are the same. We only need to do the canonical identification of $T_{f(p)}\mathbb{R}$ with $\mathbb{R}$.

Def: Let $M$ be a smooth manifold with or without boundary and $p\in M$. Let ${\ell }_p$ denote the subspace of ${\mathcal{C}}^{\mathrm{\infty }}(M)$ consisting of smooth functions that vanish at $p$, and let ${\ell }_p^2$ be the subspace of ${\ell }_p$ spanned by functions of the form $fg$ for some $f,g\in {\ell }_p$.

Prop: $f\in {\ell }_p^2$ iff in any smooth local coordinates, its first first-order Taylor polynomial at $p$ is zero. Because of this we say that a function in ${\ell }_p^2$ is said to vanish to second order.

Prop: The map $\mathrm{\Phi }:{\ell }_p\to T_p^{\ast }M$ by setting $\mathrm{\Phi }(f):=d{f}_p$. We see that restriction $\mathrm{\Phi }$ to ${\ell }_p^2$ is zero, and that $\mathrm{\Phi }$ descends to a vector space isomorphism from ${\ell }_p/{\ell }_p^2$ to $T_p^{\ast }M$.

In some treatments of smooth manifold theory, $T_p^{\ast }M$ is defined first in this way, and the then $T_{p}M$ is defined as the dual space $({\ell }_p/{\ell }_p^2{)}^{\ast }$.

Lagrange Multipliers: Let $M$ be a smooth manifold, and let $C\subseteq M$ be an embedded submanifold that admits a global defining function $\mathrm{\Phi }:M\to {\mathbb{R}}^k$. Let $f\in {\mathcal{C}}^{\mathrm{\infty }}(M)$, and suppose $p\in C$ is a point at which $f$ attains its maximum or minimum value among points in $C$. Them there are real numbers ${\lambda }_1,\dots ,{\lambda }_k$ (called Lagrange multipliers) such that $$df_p = \sum_{n = 1}^k\lambda_n d\Phi^n|_p.$$
The proof of this relies on the fact $\{d{\mathrm{\Phi }}^1{|}_p,\dots ,d{\mathrm{\Phi }}^l{|}_p\}$ forms a basis for the annihilator of $T_{p}C$. A critical point of $f{|}_C$ must have a vanishing differential, then $d(f{|}_C{)}_p$ must be an element of the annihilator of $T_{p}C$ when seen as subspace of $T_{p}M$.

Pullback of Covector Fields

Def: Let $F:M\to N$ be a smooth map between smooth manifolds with or without boundary, and let $p\in M$ be arbitrary. The differential $d{F}_p:T_{p}M\to T_{F(p)}N$ yields a dual linear map $$dF_p^: T_{F(p)}^ N \to T_p^M,$$called the pointwise pullback by $F$ at $p$, or the cotangent map of $F$. We see that $d{F}_p^{\ast }$ is characterised by $$dF_p^(\omega)(v) := \omega(dF_p(v)), \qquad \text{for }\omega\in T^*_{F(p)} \text{ and } v\in T_p M.$$
We see that the assignments $(M,p)\mapsto T_p^{\ast }M$ and $F\mapsto d{F}_p^{\ast }$ yield a contravariant functor from the category of pointed smooth manifold to the category of real vector spaces. The convention of calling elements of $T^{\ast }M$ 'covariant vectors' is particularly unfortunate.

Def: Given a smooth map $F:M\to N$ and a covector field $\omega$ on $N$, we define the a rough covector field $F^{\ast }\omega$ on $M$, called the pullback of $\omega$ by $F$, by $$(F^\omega)_p := dF^p(\omega). $$It acts on a vector $v\in T_{p}M$ by $$(F^\omega)p(v) := \omega (dF_p(v)). $$
Prop: Let $F:M\to N$ be a smooth map between smooth manifold with or without boundary. Suppose $u$ is a continuous real-valued function on $N$, and $\omega$ is a covector field on $N$. Then $$F^(u\omega) = (u\circ F)(F^\omega). $$If in addition $u$ is smooth, then $$F^du= d(u\circ F). $$
Cor: Let $p\in M$, and choose smooth coordinates $(y^j)$ for $N$ in a nieghbourhood $V$ of $F(p).$ Let $U:=F^{-1}[V]$, which is a neighbourhood of $p$. We can write a covector field $\omega$ in coordinates as $\omega ={\omega }_{j}d{y}^j$, then $$F^\omega =F^(\omega_j dy^j) = (\omega_j \circ F) F^*dy^j = (\omega_j\circ F) d(y^j \circ F) = (\omega_j\circ F)dF^j.$$

Prop: Suppose $F:M\to N$ is a smooth map between manifold with or boundary, and let $\omega$ be a covector field on $N$. Then $F^{\ast }\omega$ is a (continuous) covector field on $M$. If $\omega$ is smooth, then so is $F^{\ast }\omega$.

Prop: Suppose $F:M\to N$ is a diffeomorphism, and let $d{F}^{\ast }:T^{\ast }N\to T^{\ast }M$ be the map whose restriction to each cotangent space $T_q^{\ast }N$ is equal to $d{F}_{F^{-1}(q)}^{\ast }$, then $d{F}^{\ast }$ is a smooth bundle homomorphism.

Prop: Let $\mathsf{D}\mathsf{i}\mathsf{f}{\mathsf{f}}_{\mathsf{1}}$ be the category whose objects are smooth manifolds, but whose only morphisms are diffeomorphism; and let $\mathsf{V}\mathsf{B}$ be the category whose objects are smooth vector bundles and whose morphisms are smooth vector bundle homomorphisms. Then, the assignment $M\mapsto T^{\ast }M$, $F\mapsto d{F}^{\ast }$ defined a contravariant functor from $\mathsf{D}\mathsf{i}\mathsf{f}{\mathsf{f}}_{\mathsf{1}}$ to $\mathsf{V}\mathsf{B}$ called the cotangent functor.

Restricting Covector Fields on Submanifold

Suppose $M$ is a smooth manifold with or without boundary, $S\subseteq M$ is an immersed submanifold with or without manifold, and let $\iota :S\hookrightarrow M$ is the inclusion map. If $\omega$ is any smooth covector field on $M$, the pullback $\iota$ yields a smooth covector field ${\iota }^{\ast }\omega$ on $S$. Let $v\in T_{p}S$ be arbitrary, and compute $$(\iota^*\omega)_p(v) = \omega_p(d\iota_p(v)) = \omega_p(v), $$since $d{\iota }_p:T_{p}S\to T_{p}M$ is just the inclusion map. Thus, ${\iota }^{\ast }\omega$ is just the restriction of $\omega$ to vectors tangent to $S$. For this reason, ${\iota }^{\ast }\omega$ is often called the restriction of $\omega$ of $S$. We see that ${\iota }^{\ast }\omega$ might be zero at a given point of $S$, even though considered as a covector gield on $M$, $\omega$ might not vanish.

To distinguish the two ways in which we might interpret the statement '$\omega$ vanishes on $S$', one usually says that $\omega$ vanishes along $S$ or vanishes at points of $S$ if ${\omega }_p=0$ for every point $p\in S$. The weaker condition that ${\iota }^{\ast }\omega =0$ is expressed by saying the restriction of $\omega$ to $S$ vanishes, or the pullback of $\omega$ to $S$ vanishes.