These notes are a quite terse introduction to vector fields and 1-forms on
\(\mathbb{R}^n\). I will fill in missing details and answer questions
during the lecture. For more information, see one of the differential
topology references on the
References page. Specifically,
Phil Parker's
book-in-progress is a good place to start.
Let \(U\) be an open set in \(\mathbb{R}^n\) containing the origin, and denote
the coordinates on \(U\) by \(x = (x^1,x^2,...,x^n)\). We shall refer to the
pair \((U,x)\) as a chart centered at the origin in \(\mathbb{R}^n\).
Denote the set \(C^\infty(U,\mathbb{R})\) of all smooth real-valued functions on
\(U\) by \(\mathfrak{F}(U)\), or just \(\mathfrak{F}\) when \(U\) is understood
from context. The set \(\mathfrak{F}\) forms an \(\mathbb{R}\)-algebra under
pointwise operations: addition, scalar multiplication, and function multiplication.
A derivation of \(\mathfrak{F}=\mathfrak{F}(U)\) is a map \(D:
\mathfrak{F} \to \mathfrak{F}\) satisfying
| 1. |
\(D(\alpha f + \beta g) = \alpha D(f) + \beta D(g)\), and |
| 2. |
\(D(fg) = D(f)g + fD(g)\) |
for all \(f,g \in \mathfrak{F}\), and all \(\alpha, \beta \in \mathbb{R}\).
The space of all derivations on \(\mathfrak{F}\) is denoted by
\(\mathrm{Der}(\mathfrak{F})\).
The space \(\mathrm{Der}(\mathfrak{F})\) forms a
Lie algebra. Specifically,
\(\mathrm{Der}(\mathfrak{F})\) is an \(n\)-dimensional \(\mathbb{R}\)-vector
space with a bilinear product \( [\ ,\, ] : \mathrm{Der}(\mathfrak{F}) \times
\mathrm{Der}(\mathfrak{F}) \to \mathrm{Der}(\mathfrak{F})\) satisfying
| 1. |
For every \(D_1, D_2 \in \mathrm{Der}(\mathfrak{F})\),
\(
\left[D_1,D_2\right] = D_1 D_2 - D_2 D_1
\)
is also in \(\mathrm{Der}(\mathfrak{F})\); and
|
| 2. |
\(
\left[\left[D_1,D_2\right],D_3\right] + \left[\left[D_2,D_3\right],D_1\right] +
\left[\left[D_3,D_1\right],D_2\right] = 0
\)
for all \(D_1, D_2, D_3 \in \mathrm{Der}(\mathfrak{F})\).
|
The directional derivatives in the directions of the coordinates
(partial derivatives) \[\left\{\dfrac{\partial}{\partial x^1},
\dfrac{\partial}{\partial x^2},...,\dfrac{\partial}{\partial x^n}\right\} =:
\left\{\partial_1,\partial_2,...,\partial_n \right\} \tag{\(*\)}\] form a basis
of \(\mathrm{Der}(\mathfrak{F})\). A
vector field \(X\) on \(U\) is a
linear combination
\[
X = X^1\partial_1 + X^2\partial_2 + \cdots + X^n\partial_n =
\sum_{i=1}^n X^i\partial_i
\]
where \(X^i \in \mathfrak{F}\) for \(i = 1,...,n\).
It is common to employ Einstein's summation convention: when an index repeats in
a sub- and superscript of a product, then it is understood that one should sum
over that index. Using this convention, a vector field \(X\) may be written as
\(X = X^i\partial_i\).
The space of all vector fields on \(U\) is denoted by \(\mathfrak{X}(U)\), or
again just \(\mathfrak{X}\) when \(U\) is fixed or understood from context. The
space \(\mathfrak{X}\) is a Lie algebra under the commutator bracket defined
above (for derivations). The elements of \(\mathfrak{X}\) act on \(\mathfrak{F}\)
by
\[
(Xf)(p) = \sum_{i=1}^n X^i(p)\dfrac{\partial f}{\partial x^i}(p),
\]
where \( p \in U\). At any point \(p \in U \), \( X(p) =: X_p =
X^i(p)\partial_i\) is a
tangent vector to \(U\) at \(p\). The space of all
tangent vectors to \(U\) at \(p\) is \(T_pU := \mathrm{Der}_p\mathfrak{F}\), and
the disjoint union of all tangent spaces, \(TU = \sqcup_{p \in U} T_pU\), is
called the
tangent bundle of \(U\). The individual tangent spaces are
called the
fibers of the tangent bundle.
Since the tangent spaces \(T_pU\) are finite-dimensional vector spaces, one may
consider their dual spaces, \( (T_pU)^* =: T^*_p U \). The elements of \(T^*_pU\)
are linear functionals on \(T_pU\). That is, \(\varphi \in T^*_p U\) is an
\(\mathbb{R}\)-linear map \(\varphi: T_pU \to \mathbb{R}\). The space \(T^*_pU\)
is the
cotangent space to \(U\) at \(p,\) and the disjoint union
\(T^*U = \sqcup_{p \in U} T^*_pU\) is the
cotangent bundle to \(U\).
Each cotangent space \(T_p^*U\) is an \(n\)-dimensional vector space isomorphic
to \(T_pU\). Define a basis \(\left\{dx^1, dx^2,..., dx^n\right\} \)of \(T^*_pU\)
via \((*)\) by demanding that \[ dx^i(\partial_j) = \delta^i_j \] for
\(i,j = 1,...,n\), where \(\delta^i_j\) is the Kronecker delta. Elements of
\(T^*_pU\) are called
covectors, and can be represented by an
\(\mathbb{R}\)-linear combination
\[
\varphi = \phi_1 dx^1 + \phi_2 dx^2 + \cdots + \phi_n dx^n = \phi_idx^i.
\]
The
one-forms (or 1-forms) on \(U\) are \(\mathfrak{F}\)-linear combinations
\[
\theta = \theta_1 dx^1 + \theta_2 dx^2 + \cdots + \theta^n dx^n = \theta_i dx^i.
\]
One-forms act on \(\mathfrak{X}\) by
\[
\theta(X) = \theta_1X^1 dx^1(\partial_1) + \theta_2X^2 dx^2(\partial_2) + \cdots
+ \theta_nX^n dx^n(\partial_n) = \theta_iX^i \in \mathfrak{F}.
\]
The space of all 1-forms on \(U\) is denoted by \(\Omega^1(U)\) or just
\(\Omega^1\).