Differential forms, integration and the generalized Stokes theorem (part I)
If you've read the previous couple of posts, you were certainly struck by how similar the Gauss, Green and Stokes theorems are. They are all particular instances of the generalized Stokes theorem which we discuss today, in the language of differential forms. First, let me suggest a couple references:
So let's get started. One essential feature of the integrals in the previous posts was that they kept track of orientation -- this led to the internal sides canceling out and leaving us only with boundary terms. A differential form is exactly what generalizes this to arbitrary manifolds: a $k$-form is an object you can integrate over oriented $k$-dimensional manifolds, with opposite orientations leading to opposite signs.
To see how to implement this, imagine subdividing the $k$-dimensional manifold $\Sigma$ into many small parallelograms (we were using triangles previously, but parallelograms work out nicer; the geometric idea is the same). As in the proof of Stokes's theorem, each parallelogram is nearly flat, and so can be thought of as being spanned by $k$ tangent vectors $v_1, \ldots, v_k$ at a point $p$. A differential form, in a sense, gives a notion of signed 'volume' for this parallelogram. More specifically,
Definition: A (differential) $k$-form on a manifold $M$ (of arbitrary dimension) is a function $\omega$ which maps an ordered list of $k$ tangent vectors $v_1, \ldots, v_k$ (based at the same point $p$) to a real number $\omega(v_1, \ldots, v_k)$. It is required to be linear in each argument (i.e. multilinear), and alternating[1]: \[ \omega(v_1, \ldots, v_i, \ldots, v_j, \ldots, v_k) = - \omega(v_1, \ldots, v_j, \ldots, v_i, \ldots, v_k) \] for all $1 \leq i < j \leq k$. Finally, it is required to vary smoothly with $p$ (the meaning of this will become clear later).
In other words, a (differential) $k$-form on $M$ is an assignment of a linear $k$-form $\omega_p$ for each tangent space $T_p M$, where by a linear $k$-form on a vector space $V$ I just mean a multilinear and alternating map $V^k \to \mathbb{R}$.
Let's look at some examples of linear forms. We take $V = \mathbb{R}^3$, with the standard basis $e_1, e_2, e_3$. The easiest example is the determinant map (also called the triple product) \[ \det : \mathbb{R}^3 \times \mathbb{R}^3 \times \mathbb{R}^3 \to \mathbb{R} \] In the language of differential forms it is the $3$-form known as $dx \wedge dy \wedge dz$; it computes the signed volume of a given oriented parallelepiped. We can also have $2$-forms in $\mathbb{R}^3$. For instance, there's the $2$-form $dx \wedge dy$, which takes two input vectors, projects them to the $xy$ plane and computes the signed area of the parallelogram they span. There's also $dy \wedge dz$ and $dx \wedge dz$; plus, we could take linear combinations of those. It turns out that, in general, all examples are obtained this way:
Fact: Consider Euclidean space $\mathbb{R}^n$ with coordinates $x^1, \ldots, x^n$. For each increasing sequence $i_1 < \cdots < i_k$ of indices consider the linear $k$-form $dx^{i_1} \wedge \cdots \wedge dx^{i_k}$ defined by \[ (dx^{i_1} \wedge \cdots \wedge dx^{i_k})(v_1, \ldots, v_k) = \begin{vmatrix} dx^{i_1}(v_1) & \cdots & dx^{i_1}(v_k) \\ \cdots & \cdots & \cdots \\ dx^{i_k}(v_1) & \cdots & dx^{i_k}(v_k) \end{vmatrix} \] where $dx^j(v)$ is the $j$th component of $v$[2]. The collection of all such $k$-forms is a basis for the space of linear $k$-forms on $\mathbb{R}^n$.
Geometrically, the form $dx^{i_1} \wedge \cdots \wedge dx^{i_k}$ can be seen as computing the signed volume of the projection of a given oriented $k$-parallelogram onto the subspace generated by $e_{i_1}, \ldots, e_{i_k}$. Although I've been writing the $\wedge$ (wedge) symbol as a mere notational device, you can now see how it defines a product of forms. For example, \[ (dx^1 + 2dx^3) \wedge (dx^2 \wedge dx^4) = dx^1 \wedge dx^2 \wedge dx^4 - 2 dx^2 \wedge dx^3 \wedge dx^4 \] where I used the rule $dx^3 \wedge dx^2 = -dx^2 \wedge dx^3$ (analogous to how swapping two rows in a determinant flips its sign). It's possible to give a general and abstract definition of $\wedge$ purely in terms of multilinear alternating maps, but I don't find it at all enlightening so I will skip it.
Now that we understand linear forms, let's return to differential forms. The tangent space at each point of $\mathbb{R}^n$ is just $\mathbb{R}^n$ itself. Since a differential form assigns to each point a linear form on the corresponding tangent space, a differential form on $\mathbb{R}^n$ can also be written as a linear combination of the basic forms $dx^{i_1} \wedge \cdots \wedge dx^{i_n}$, except the coefficients are allowed to depend on the point $p$. Therefore, the general expression for a $k$-form on $\mathbb{R}^n$ is[3] \[ \omega_p = \sum_{i_1 < \cdots < i_k} f_{i_1, \ldots, i_k}(p) dx^{i_1} \wedge \cdots \wedge dx^{i_k} \] The smoothness of $\omega$ is dictated by the smoothness of the coefficient functions $f_{i_1, \ldots, i_k}$. They are usually taken to be $C^{\infty}$, although for Stokes's theorem $C^2$ is enough.
Hopefully this post has motivated why differential forms are the correct objects to generalize oriented integration -- we'll see how to do that next time.
- Hubbard & Hubbard - Vector Calculus, Linear Algebra and Differential Forms: A Unified Approach. This is an outstanding book on multivariable calculus, with the best presentation of differential forms I've seen anywhere. It has detailed proofs of the ideas sketched in this series of posts.
- Terence Tao - Differential Forms and Integration. Lovely article which really helped me gain some intuition about differential forms.
- Baez & Munian - Gauge Fields, Knots and Gravity. Don't be misled by the physics title -- this book has one of the clearest expositions of basic differential geometry I've seen (rivaled perhaps by Spivak's books).
To see how to implement this, imagine subdividing the $k$-dimensional manifold $\Sigma$ into many small parallelograms (we were using triangles previously, but parallelograms work out nicer; the geometric idea is the same). As in the proof of Stokes's theorem, each parallelogram is nearly flat, and so can be thought of as being spanned by $k$ tangent vectors $v_1, \ldots, v_k$ at a point $p$. A differential form, in a sense, gives a notion of signed 'volume' for this parallelogram. More specifically,
Definition: A (differential) $k$-form on a manifold $M$ (of arbitrary dimension) is a function $\omega$ which maps an ordered list of $k$ tangent vectors $v_1, \ldots, v_k$ (based at the same point $p$) to a real number $\omega(v_1, \ldots, v_k)$. It is required to be linear in each argument (i.e. multilinear), and alternating[1]: \[ \omega(v_1, \ldots, v_i, \ldots, v_j, \ldots, v_k) = - \omega(v_1, \ldots, v_j, \ldots, v_i, \ldots, v_k) \] for all $1 \leq i < j \leq k$. Finally, it is required to vary smoothly with $p$ (the meaning of this will become clear later).
In other words, a (differential) $k$-form on $M$ is an assignment of a linear $k$-form $\omega_p$ for each tangent space $T_p M$, where by a linear $k$-form on a vector space $V$ I just mean a multilinear and alternating map $V^k \to \mathbb{R}$.
Let's look at some examples of linear forms. We take $V = \mathbb{R}^3$, with the standard basis $e_1, e_2, e_3$. The easiest example is the determinant map (also called the triple product) \[ \det : \mathbb{R}^3 \times \mathbb{R}^3 \times \mathbb{R}^3 \to \mathbb{R} \] In the language of differential forms it is the $3$-form known as $dx \wedge dy \wedge dz$; it computes the signed volume of a given oriented parallelepiped. We can also have $2$-forms in $\mathbb{R}^3$. For instance, there's the $2$-form $dx \wedge dy$, which takes two input vectors, projects them to the $xy$ plane and computes the signed area of the parallelogram they span. There's also $dy \wedge dz$ and $dx \wedge dz$; plus, we could take linear combinations of those. It turns out that, in general, all examples are obtained this way:
Fact: Consider Euclidean space $\mathbb{R}^n$ with coordinates $x^1, \ldots, x^n$. For each increasing sequence $i_1 < \cdots < i_k$ of indices consider the linear $k$-form $dx^{i_1} \wedge \cdots \wedge dx^{i_k}$ defined by \[ (dx^{i_1} \wedge \cdots \wedge dx^{i_k})(v_1, \ldots, v_k) = \begin{vmatrix} dx^{i_1}(v_1) & \cdots & dx^{i_1}(v_k) \\ \cdots & \cdots & \cdots \\ dx^{i_k}(v_1) & \cdots & dx^{i_k}(v_k) \end{vmatrix} \] where $dx^j(v)$ is the $j$th component of $v$[2]. The collection of all such $k$-forms is a basis for the space of linear $k$-forms on $\mathbb{R}^n$.
Geometrically, the form $dx^{i_1} \wedge \cdots \wedge dx^{i_k}$ can be seen as computing the signed volume of the projection of a given oriented $k$-parallelogram onto the subspace generated by $e_{i_1}, \ldots, e_{i_k}$. Although I've been writing the $\wedge$ (wedge) symbol as a mere notational device, you can now see how it defines a product of forms. For example, \[ (dx^1 + 2dx^3) \wedge (dx^2 \wedge dx^4) = dx^1 \wedge dx^2 \wedge dx^4 - 2 dx^2 \wedge dx^3 \wedge dx^4 \] where I used the rule $dx^3 \wedge dx^2 = -dx^2 \wedge dx^3$ (analogous to how swapping two rows in a determinant flips its sign). It's possible to give a general and abstract definition of $\wedge$ purely in terms of multilinear alternating maps, but I don't find it at all enlightening so I will skip it.
Now that we understand linear forms, let's return to differential forms. The tangent space at each point of $\mathbb{R}^n$ is just $\mathbb{R}^n$ itself. Since a differential form assigns to each point a linear form on the corresponding tangent space, a differential form on $\mathbb{R}^n$ can also be written as a linear combination of the basic forms $dx^{i_1} \wedge \cdots \wedge dx^{i_n}$, except the coefficients are allowed to depend on the point $p$. Therefore, the general expression for a $k$-form on $\mathbb{R}^n$ is[3] \[ \omega_p = \sum_{i_1 < \cdots < i_k} f_{i_1, \ldots, i_k}(p) dx^{i_1} \wedge \cdots \wedge dx^{i_k} \] The smoothness of $\omega$ is dictated by the smoothness of the coefficient functions $f_{i_1, \ldots, i_k}$. They are usually taken to be $C^{\infty}$, although for Stokes's theorem $C^2$ is enough.
Hopefully this post has motivated why differential forms are the correct objects to generalize oriented integration -- we'll see how to do that next time.
Notes
[1] Compare this to the fact that, for an $n$-dimensional vector space $V$, any multilinear alternating function $V^n \to \mathbb{R}$ is a constant multiple of the determinant (with respect to some basis). ↩
[2] Thus $dx^j$ is a linear $1$-form. ↩
[3] In a general $n$-dimensional manifold, we may always locally choose coordinates $x^1, \ldots, x^n$ and write a $k$-form as the given expression.↩
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