We summarize here the basic properties of differential forms. See Chapter 12 for further details.

Wedge Products

Differential forms are integrands, the things one integrates. So $dx$ is a differential form (a 1-form), and so is $dx\,dy$ (a 2-form). However, orientation matters; think about change of variables, where for instance \begin{equation} du\,dv = \Jacobian{u}{v}{x}{y} \> dx\,dy \end{equation} where $\Jacobian{u}{v}{x}{y}$ denotes the determinant of the Jacobian matrix of $(u,v)$ with respect to $(x,y)$. One often puts an absolute value sign around this determinant, but that's misleading. For example, flux depends on the orientation of the surface, so it's a feature, not a bug, for integrals to depend on the ordering (“handedness”) of the coordinates.

In order to emphasize that order matters, we write the exterior product of differential forms using the symbol $\wedge$, which is read as “wedge”. The above formula becomes \begin{equation} du \wedge dv = \Jacobian{u}{v}{x}{y} \> dx \wedge dy \end{equation} from which we see that \begin{align} dy \wedge dx &= - dx \wedge dy \\ dx \wedge dx &= 0 \end{align}

So order matters. But parentheses are not needed; the wedge product is associative (so $dx\wedge dy\wedge dz$ is unambiguous).

It is worth noting that, by construction, differential forms automatically incorporate change of variables. For example, comparing rectangular and polar coordinates, we have \begin{equation} dx \wedge dy = r\,dr \wedge d\phi \end{equation} an equality which does not in fact hold for differentials, despite frequent usage to the contrary. 1)

Given coordinates $(x^i)$ on an $n$-dimensional space $M$, it is easy to see that any 1-form can be written as \begin{equation} \alpha = \alpha_i \> dx^i \end{equation} where we have adopted the Einstein summation convention in which a sum is implied by the presence of repeated indices. In other words, $\{dx^i\}$ forms a basis for 1-forms. Similarly, $\{dx^i\wedge dx^j\}$ with $i<j$ is a basis for 2-forms, and so forth. We also consider scalars (functions) to be 0-forms, and if necessary interpret wedge products of functions as ordinary multiplication.

For further insight into the meaning of $\wedge$, the reader is encouraged to work out $\alpha\wedge\beta$ for two arbitrary 1-forms in 3-dimensional Euclidean space.

Orthonormal Frames

We will always work in an orthonormal basis. Intuitively, this means that the basis 1-forms measure “infinitesimal distance” in mutually orthogonal directions. Two such bases in 2-dimensional Euclidean space are $\{dx,dy\}$ and $\{dr,r\,d\phi\}$, which can easily be remembered using either the infinitesimal vector displacement \begin{equation} d\rr = dx\,\xhat + dy\,\yhat = dr\,\rhat + r\,d\phi\,\phat \end{equation} or the line element (also called the metric) \begin{equation} ds^2 = d\rr\cdot d\rr = dx^2 + dy^2 = dr^2 + r^2\,d\phi^2 \end{equation} In general, we will write our orthonormal basis of 1-forms as $\{\sigma^i\}$, so that a generic 1-form can be expanded as \begin{equation} \beta = \beta_i \> \sigma^i \end{equation}

We introduce the notation $g(\alpha,\beta)$ for the “dot product” of two 1-forms, which is easy to compute in terms of the multiplication table for an orthonormal basis. However, we do not require the dot product to be positive definite; in particular, the “squared magnitude” of the elements in our basis can be either positive or negative. For example, in two-dimensional Minkowski space, which we have been calling hyperbola geometry and which describes special relativity, we have \begin{equation} d\rr = dt\,\That + dx\,\xhat \end{equation} and \begin{equation} ds^2 = d\rr\cdot d\rr = -dt^2 + dx^2 \end{equation} since $\That\cdot\That=-1$. In this geometry, we have $g(dx,dx)=1$, but $g(dt,dt)=-1$.

It is straightforward to extend the inner product $g$ to higher-rank differential forms, although one must be careful with signs.

Hodge Dual

Given an orthonormal basis in $n$ dimensions, there are exactly two choices of volume element, that is, of a unit $n$-form, obtained by multiplying together the basis 1-forms in any order, since two such products are the same if the orderings differ by an even permutation. Choosing one of these volume elements fixes an orientation. For example, in two Euclidean dimensions, the standard orientation is given by \begin{equation} \omega = dx \wedge dy = dr \wedge r\,d\phi \end{equation}

Given any $p$-form, there is a natural ($n-p$)-form associated with it, which consists roughly of the “missing pieces” needed to make up the (given) volume element $\omega$. We write $*\alpha$ for the differential form associated in this way with $\alpha$, which is called the Hodge dual of $\alpha$. Order matters, so $*dx=dy$, but $*dy=-dx$. In general, \begin{equation} \alpha\wedge*\beta = g(\alpha,\beta) \, \omega \end{equation} for any $p$-forms $\alpha$, $\beta$, and this can be used to work out $*\beta$.

An important property of the Hodge dual is that \begin{equation} ** = (-1)^{p(n-p)+s} \end{equation} where $p$ is the rank of the differential form being acted on, and $s$ is the signature of the metric, that is, the number of basis elements with negative squared magnitude.

For further insight into the meaning of $*$, the reader is encouraged to work out ${*}(\alpha\wedge{*}\beta)$ for two arbitrary 1-forms in 3-dimensional Euclidean space.

Exterior Differentiation

Not only do we integrate differential forms (in the obvious way); we also differentiate them. We already know how to differentiate 0-forms, namely \begin{equation} d(f) = df = \Partial{f}{x^i} \> dx^i \end{equation} We can generalize this operation to higher rank forms by requiring \begin{equation} d(f \, dx \wedge … \wedge dy) = df \wedge dx \wedge … \wedge dy \end{equation} from which it follows that \begin{align} d^2 &= 0 \\ d(\alpha \wedge \beta) &= d\alpha \wedge \beta + (-1)^p\, \alpha \wedge d\beta \end{align} where $\alpha$ is a $p$-form.

For further insight into the meaning of $d$, the reader is encouraged to work out $*d\alpha$ and $*d{*}\beta$ for two arbitrary 1-forms in 3-dimensional Euclidean space.

Connections and Curvature

The connection 1-forms describe how the basis changes. In an orthonormal frame with \begin{equation} d\rr = \sigma^i\,\ee_i \end{equation} the connection 1-forms satisfy \begin{align} d\ee_j &= \omega^i{}_j\,\ee_i \\ \omega_{ij} &= \ee_i \cdot d\ee_j \end{align} where each $\omega^i{}_j$ and $\omega_{ij}$ differ at most by a sign, depending on the metric signature. We always work with the Levi-Civita connection, that is, we make the assumptions that the connection is metric-compatible \begin{equation} d(\vv\cdot\ww) = d\vv\cdot\ww + \vv\cdot d\ww \end{equation} and torsion-free \begin{equation} d^2\rr = d(d\rr) = 0 \end{equation} For a given line element ($ds^2$; it is enough to give $d\rr$), there is a unique connection satisfying these conditions, which is the unique solution to the system of equations \begin{align} d\sigma^i + \omega^i{}_j \wedge \sigma^j &= 0 \\ \omega_{ij} + \omega_{ji} &= 0 \end{align} where again the “up” and “down” indices reflect the signature, and incorporate a factor of $-1$ for “negatively normed” basis elements.

The curvature 2-forms describe the shape of the given space, and are defined by \begin{equation} d^2 \ee_j = \Omega^i{}_j \ee_i \end{equation} which turns out to imply \begin{equation} \Omega^i{}_j = d\omega^i{}_j + \omega^i{}_m \wedge \omega^m{}_j \end{equation}

Thus, if \begin{equation} \vv = v^i \ee_i \end{equation} then \begin{equation} d\vv = dv^i \ee_i + v^i d\ee_i = \left(dv^i + v^j \omega^i{}_j\right) d\ee_i \end{equation} and \begin{equation} d^2\vv = v^i d^2\ee_i = v^i \Omega^j{}_i \ee_j \end{equation}

Finally, we introduce the notation 2) \begin{align} \omega^i{}_j &= \Gamma^i{}_{jk}\,\sigma^k \\ \Omega^i{}_j &= \frac12 R^i{}_{jkm}\,\sigma^k\wedge\sigma^m \label{RiemNut} \end{align} for the components of the connection and curvature forms; the functions $\Gamma^i{}_{jk}$ are usually referred to as Christoffel symbols, and the $R^i{}_{jkm}$ are the components of the Riemann curvature tensor.

For further insight into the meaning of connections and curvature, the reader is encouraged to compute these forms in 2-dimensional Euclidean space in both rectangular and polar coordinates.

See Part III for further details.