Integration and Stokes’ Theorem

Abstract

For a curve $\gamma \colon [a,b]\to \mathbb {R}^n$, global quantities like the length or the bending energy were defined as integrals over arclength of certain functions on $[a,b]$.

For a curve $\gamma \colon [a,b]\to \mathbb {R}^n$, global quantities like the length or the bending energy were defined as integrals over arclength of certain functions on $[a,b]$. Before we can define similar quantities for a surface $f \colon M\to \mathbb {R}^n$, for example, the area of f, we have to find a way to integrate functions $g \colon M\to \mathbb {R}$ in a geometrically meaningful fashion. We will do this in terms of the area 2-form$\det$ of the metric induced by f. This means that first we have to develop the theory of differential forms on two-dimensional domains M, including the theorem of Stokes.

1 Integration on Surfaces

Let $M,\tilde {M}\subset \mathbb {R}^2$ be two compact domains with smooth boundary, $\varphi \colon \tilde {M} \to M$ an orientation-preserving diffeomorphism and $g\colon M\to \mathbb {R}$ a smooth function. By the transformation formula for integrals, we have

$$\displaystyle \begin{aligned} \int_{\tilde{M}} g\circ \varphi \,\det \varphi' = \int_M g.\end{aligned}$$

Therefore, if one were to just use $\int _Mg$ as the definition for an integral of a function g over the surface f, this integral would not be invariant under reparametrization of f. On the other hand, we are now going to convince ourselves that it is perfectly possible to define the integral of an object like the area form $\det$, as it was introduced in Theorem 6.22:

Definition 7.1

Let M be a compact domain with smooth boundary in $\mathbb {R}^2$. Then a map

$$\displaystyle \begin{aligned} \sigma\colon \bigcup_{p\in M} (T_pM\times T_pM) \to \mathbb{R}\end{aligned}$$

is called a 2-form on M if for each $p\in M$ the restriction of $\sigma$ to $T_pM\times T_pM$ is a skew-symmetric bilinear form and the function

$$\displaystyle \begin{aligned} \sigma(U,V)\colon M\to\mathbb{R}\end{aligned}$$

is smooth.

We denote the set of all 2-forms on M by $\Omega ^2(M)$. As a linear subspace of the vector space of all real-valued functions on some set, also $\Omega ^2(M)$ is a real vector space. It is also clear how to define the product of a 2-form $\sigma$ with a function $g\in C^\infty (M)$. One can say that 2-forms are similar to Riemannian metrics, only skew symmetric instead of symmetric and without any non-degeneracy assumptions.

Remark 7.2

A “model example” of a 2-form which we have already encountered is the well-known $\det \in \Omega ^2(M)$.

2-forms are transported under diffeomorphisms by demanding that the transported form applied to the transported tangent vectors yields the same value as before:

Definition 7.3

Let $M,\tilde {M}\subset \mathbb {R}^2$ be two compact domains with smooth boundary, $\varphi \colon \tilde {M}\to M$ a smooth map and $\sigma$ a 2-form on M. Then we define the pull-back of $\sigma$ under $\varphi$ as the 2-form $\varphi ^*\sigma$ on $\tilde {M}$ that for $p\in \tilde {M}$ and $X,Y\in T_p\tilde {M}$ is given by

$$\displaystyle \begin{aligned} (\varphi^*\sigma)(X,Y):=\sigma(d\varphi(X),d\varphi(Y)).\end{aligned}$$

Theorem 7.4

In the situation of Definition7.3, the map

$$\displaystyle \begin{aligned} \Omega^2(M) &\to \Omega^2(\tilde{M}),\ \sigma \mapsto \varphi^*\sigma\end{aligned} $$

is linear and for$g\in C^\infty (M)$satisfies

$$\displaystyle \begin{aligned} \varphi^*(g\sigma) = (g\circ \varphi) (\varphi^*\sigma).\end{aligned}$$

The integral over M of a 2-form $\sigma \in \Omega ^2(M)$ is defined as follows:

Definition 7.5

The integralof a 2-form$\sigma$ on a compact domain M with smooth boundary in $\mathbb {R}^2$ is defined as

$$\displaystyle \begin{aligned} \int_M\sigma = \int_M \sigma(U,V)\end{aligned}$$

where U and V  are the two vector fields on M introduced in Definition 6.10.

The above definition is useful because $\int _M \sigma$ is invariant under pull-back of $\sigma$ by an orientation-preserving diffeomorphism $\varphi \colon \tilde {M}\to M$.

Theorem 7.6

Let$M,\tilde {M}\subset \mathbb {R}^2$be two compact domains with smooth boundary,$\varphi \colon \tilde {M} \to M$an orientation-preserving diffeomorphism and$\sigma \in \Omega ^2(M)$a 2-form. Then

$$\displaystyle \begin{aligned} \int_{\tilde{M}}\varphi^*\sigma =\int_M \sigma.\end{aligned}$$

Proof

Let us write

$$\displaystyle \begin{aligned} \varphi'=\begin{pmatrix}a & b \\ c & d \end{pmatrix},\end{aligned}$$

which means

$$\displaystyle \begin{aligned} d\varphi(\tilde{U})=a \, U\circ \varphi + c\, V\circ \varphi \\ d\varphi(\tilde{V})=b \, U\circ \varphi + d\, V\circ \varphi.\end{aligned} $$

Therefore, by the skew symmetry of $\sigma$ and the transformation formula,

$$\displaystyle \begin{aligned} \int_{\tilde{M}}\varphi^*\sigma &= \int_{\tilde{M}} (\varphi^*\sigma)(\tilde{U},\tilde{V}) \\ &= \int_{\tilde{M}} \sigma(d\varphi(\tilde{U}),d\varphi(\tilde{V}))\\ &= \int_{\tilde{M}} (ad-bc) \,\sigma(U,V)\circ \varphi \\ &=\int_M\sigma(U,V) \\ &=\int_M \sigma.\end{aligned} $$

□

In the context of surfaces $f\colon M\to \mathbb {R}^n$, we will never integrate functions $g\in C^\infty (M)$ directly, but instead we will first make g into a 2-form by multiplying it with the area form $\det$ of the induced metric. Then we can be sure that

$$\displaystyle \begin{aligned} \int_M g\det\end{aligned}$$

is a quantity that will stay the same if we reparametrize f as $\tilde {f}=f\circ \varphi$ (and, of course, simultaneously change g to $g\circ \varphi$). Theorem 7.6 above makes it possible to define the area of a Riemannian domain in such a way that it does not change under isometries:

Definition 7.7

The area of a Riemannian domain$(M,\langle \,,\rangle )$ is defined as

$$\displaystyle \begin{aligned} \int_M \det\end{aligned}$$

where $\det$ is the area form of $\langle \,,\rangle$.

2 Integration Over Curves

In order to adequately deal with surfaces, we have found it necessary to add tangent spaces, Riemannian metrics and 2-forms to our toolbox. Let us investigate whether some of these notions might be useful already in the context of curves. For a curve $\gamma \colon [a,b]\to \mathbb {R}$, the analog of the domain M of a surface $f\colon M\to \mathbb {R}^n$ is the interval $[a,b]$. The tangent bundle of $[a,b]$ is

$$\displaystyle \begin{aligned} T[a,b]=[a,b]\times \mathbb{R}\end{aligned}$$

and the tangent space at $p\in [a,b]$ is $\{p\}\times \mathbb {R}$. The analog of the vector fields $U,V$ on M is the single vector field $X\in \Gamma ([a,b])$ defined as

$$\displaystyle \begin{aligned} X(p)=(p,1).\end{aligned}$$

The objects that can naturally be integrated over curves are the so-called 1-forms. We will need 1-forms also on planar domains (Fig. 7.1), so we take the opportunity to define also those.

Fig. 7.1

A 1-form $\omega \in \Omega ^1(M)$ can be thought of as a smoothly varying ruler which “measures” a vector field $X\in \Gamma (M)$. The spacing of the ruler-lines indicates the “strength” of $\omega$—the closer the spacing, the stronger is $\omega$

Definition 7.8

Let $[a, b]$ be a closed interval and $M\subset \mathbb {R}^2$ a planar domain with smooth boundary. Smoothness of maps defined on $T{[a,b]}$ or TM is to be understood in the sense of Definition A.1. Then

• (i) A smooth map $\omega \colon T[a,b]\to \mathbb {R}$ is called a 1-form if its restriction to each tangent space $T_p[a,b]$ is linear. The space of all 1-forms on $[a,b]$ is denoted by $\Omega ^1([a,b])$.

• (ii) A smooth map $\omega \colon TM\to \mathbb {R}$ is called a 1-form if its restriction to each tangent space $T_pM$ is linear. The space of all 1-forms on M is denoted by $\Omega ^1(M)$.

A general theory of m-forms on domains in $\mathbb {R}^k$ is beyond the scope of this book, so we just collect some special cases that we need:

Definition 7.9

Let $[a,b]\subset \mathbb {R}$ be a closed interval and $M\subset \mathbb {R}^2$ a planar domain with smooth boundary.

• (i) If $\omega$ is a 1-form on $[a,b]$ and $\varphi \colon [\tilde {a},\tilde {b}] \to [a,b]$ is a smooth map, then we define the pull-back of $\omega$ under $\varphi$ as the 1-form $\varphi ^*\omega \in \Omega ^1([\tilde {a},\tilde {b}])$ which is for all $Y\in \Gamma (T[\tilde a,\tilde b])$ given by

  $$\displaystyle \begin{aligned} (\varphi^*\omega)(Y)=\omega(d\varphi(Y)).\end{aligned}$$

• (ii) If $\omega$ is a 1-form on M and $\varphi \colon \tilde {M} \to M$ is a smooth map, then we define the pull-back of $\omega$ under $\varphi$ as the 1-form $\varphi ^*\omega \in \Omega ^1(\tilde {M})$ which is for all $Y\in \Gamma (T\tilde M)$ given by

  $$\displaystyle \begin{aligned} (\varphi^*\omega)(Y)=\omega(d\varphi(Y)).\end{aligned}$$

• (iii) If $\omega$ is a 1-form on M and $\gamma \colon [a,b] \to M$ is a smooth map, then we define the pull-back of $\omega$ under $\gamma$ as the 1-form $\gamma ^*\omega \in \Omega ^1([a,b])$ which is for all $Y\in \Gamma (T[a,b])$ given by

  $$\displaystyle \begin{aligned} (\gamma^*\omega)(Y)=\omega(d\gamma(Y)).\end{aligned}$$

Definition 7.10

For $\omega \in \Omega ^1([a,b])$ we define the integralof a 1-form$\omega$ over $[a,b]$ as

$$\displaystyle \begin{aligned} \int_{[a,b]} \omega := \int_a^b \omega(X).\end{aligned}$$

Theorem 7.11

Let$\varphi \colon [\tilde {a},\tilde {b}] \to [a,b]$be an orientation-preserving diffeomorphism, i.e. a bijective smooth map with$\varphi '>0$. Then

$$\displaystyle \begin{aligned} \int_{[\tilde{a},\tilde{b}]}\varphi^*\omega = \int_{[a,b]}\omega.\end{aligned}$$

Proof

By the substitution rule and $d\varphi (\tilde {X})=\varphi ' \cdot X\circ \varphi$ we have

$$\displaystyle \begin{aligned} \int_{[\tilde{a},\tilde{b}]}\varphi^*\omega &= \int_{\tilde{a}}^{\tilde{b}} (\varphi^*\omega)(\tilde{X}) \\&= \int_{\tilde{a}}^{\tilde{b}} \omega(d\varphi(\tilde{X})) \\&= \int_{\tilde{a}}^{\tilde{b}} \omega(\varphi' \cdot X\circ \varphi)\\&= \int_{\tilde{a}}^{\tilde{b}} \varphi'\cdot \omega(X)\circ \varphi \\&= \int_a^b \omega(X) \\&= \int_{[a,b]}\omega.\end{aligned} $$

□

Theorem 7.12

Let$M\subset \mathbb {R}^2$be a compact domain with smooth boundary and$\omega \in \Omega ^1(M)$a 1-form. Let$\tilde {\gamma }\colon [\tilde {a},\tilde {b}] \to M$be a reparametrization of a smooth map$\gamma \colon [a,b]\to M$, so$\tilde {\gamma }=\gamma \circ \varphi$for an orientation-preserving diffeomorphism$\varphi \colon [\tilde {a},\tilde {b}] \to [a,b]$. Then

$$\displaystyle \begin{aligned} \int_{[\tilde{a},\tilde{b}]}\tilde{\gamma}^*\omega = \int_{[a,b]}\gamma^*\omega.\end{aligned}$$

Proof

By the chain rule, we have $d\tilde {\gamma }=d\gamma \circ d\varphi$ and therefore

$$\displaystyle \begin{aligned} \tilde{\gamma}^*\omega=\varphi^*(\gamma^*\omega).\end{aligned}$$

Therefore, Theorem 7.11 gives us

$$\displaystyle \begin{aligned} \int_{[\tilde{a},\tilde{b}]}\tilde{\gamma}^*\omega = \int_{[\tilde{a},\tilde{b}]}\varphi^*(\gamma^*\omega)=\int_{[a,b]}\gamma^*\omega.\end{aligned}$$

□

As a consequence, we can define the integral of a 1-form $\omega$ on M over a curve in M in a way that is invariant under reparametrization:

Definition 7.13

Let $M\subset \mathbb {R}^2$ be a compact domain with smooth boundary and $\omega \in \Omega ^1(M)$. Let $\gamma \colon [a,b]\to M$ be a curve. Then we define

$$\displaystyle \begin{aligned} \int_{\gamma}\omega:= \int_{[a,b]} \gamma^*\omega.\end{aligned}$$

In the context of a regular curve $\gamma \colon [a,b]\to \mathbb {R}^n$, what is the analog of the area form $\det$ of a surface $f\colon M\to \mathbb {R}^n$?

Definition 7.14

The arclength 1-form$ds\in \Omega ^1([a,b])$ of a curve $\gamma \colon [a,b]\to \mathbb {R}^n$ is defined as

$$\displaystyle \begin{aligned} ds(X)=|d\gamma(X)|=|\gamma'|.\end{aligned}$$

If we define the arclength function s as in Definition 1.13, the arclength 1-form ds is indeed the derivative of s, which explains the notation. Moreover, if we interpret the left-hand side according to Definition 7.13 and the right-hand side according to Definition 1.14, for a function $g\colon [a,b]\to \mathbb {R}$ we have

$$\displaystyle \begin{aligned} \int_{[a,b]} g\,ds =\int_a^b g\,ds.\end{aligned}$$

3 Stokes’ Theorem

When dealing with curves, we frequently used the fundamental theorem of calculus, for example in the form of integration by parts. Also in surface theory we would no get very far without the surface analog of this theorem, which is the so-called Stokes’ theorem.

Let $M\subset \mathbb {R}^2$ be a compact connected domain with smooth boundary. The boundary $\partial M$ of M can be parametrized by a finite collection of n closed curves

$$\displaystyle \begin{aligned} \gamma_j\colon [a_j,b_j]\to \mathbb{R}^2\end{aligned}$$

where $j\in \{1,\ldots n\}$. We assume that each $\gamma _j$ is oriented in such a way that for any vector $Y\in \mathbb {R}^2$ which at $\gamma _j(x)$ points out of M, we have

$$\displaystyle \begin{aligned} \det(Y,\gamma_j^{\prime}(x))>0.\end{aligned}$$

Given a 1-form $\omega \in \Omega ^1(M)$, we make use of Definition 7.13 in order to define the integral of $\omega$ over $\partial M$:

$$\displaystyle \begin{aligned} \int_{\partial M}\omega = \sum_{j=1}^n \int_{\gamma_j} \omega.\end{aligned}$$

Theorem 7.15 (Stokes Theorem)

Let$M\subset \mathbb {R}^2$be a compact domain with smooth boundary and$\omega \in \Omega ^1(M)$. Then there is a unique 2-form$d\omega \in \Omega ^2(M)$such that for all subdomains$\tilde {M}\subset M$we have

$$\displaystyle \begin{aligned} \int_{\tilde{M}} d\omega = \int_{\partial \tilde{M}}\omega.\end{aligned}$$

In fact,$d\omega$is the unique 2-form on M that satisfies

$$\displaystyle \begin{aligned} d\omega(U,V)=\omega(V)_u-\omega(U)_v.\end{aligned}$$

Proof

$\tilde {M}$ could be an arbitrarily small disk around an arbitrary point p in the interior of M, so there can be at most one 2-form $\sigma$ with the property that for all subdomains $\tilde {M}$

$$\displaystyle \begin{aligned} \int_{\tilde{M}}\sigma = \int_{\partial \tilde{M}}\omega.\end{aligned}$$

This proves the uniqueness part of the claim. If we write

$$\displaystyle \begin{aligned} \gamma_j^{\prime}=\begin{pmatrix}\alpha_j \\ \beta_j\end{pmatrix}\end{aligned}$$

we have

$$\displaystyle \begin{aligned} d\gamma_j(X)= \alpha_j \,U\circ \gamma_j + \beta_j\, V\circ \gamma_j\end{aligned}$$

and therefore

$$\displaystyle \begin{aligned} \omega(d\gamma_j(X))=\alpha_j \, \omega(U)\circ \gamma_j + \beta_j\, \omega(V) \circ \gamma_j.\end{aligned}$$

Let us define $\sigma \in \Omega ^2(M)$ as the unique 2-form for which

$$\displaystyle \begin{aligned} \sigma(U,V)= \omega(V)_u-\omega(U)_v.\end{aligned}$$

Now we apply Green’s theorem from vector calculus to the map

$$\displaystyle \begin{aligned} Y\colon M &\to \mathbb{R}^2,\ Y=\begin{pmatrix}\omega(U) \\ \omega(V)\end{pmatrix},\end{aligned} $$

and obtain

$$\displaystyle \begin{aligned} \int_M \sigma &= \int_M \sigma(U,V) \\&=\int_M (\omega(V)_u-\omega(U)_v) \\&= \sum_{j=1}^n \int_{a_j}^{b_j} (\alpha_j \, \omega(U)\circ \gamma_j + \beta_j\, \omega(V) \circ \gamma_j)\\&= \sum_{j=1}^n \int_{a_j}^{b_j} \omega(d\gamma_j(X)) \\&= \sum_{j=1}^n \int_{a_j}^{b_j} \gamma_j^*\omega \\ &=\int_{\partial M} \omega.\end{aligned} $$

We can apply this argument also to any subdomain $\tilde {M}\subset M$, which proves the existence part of the claim. □

Theorem 7.16

If$\varphi \colon \tilde {M}\to M$is a smooth map and$\omega \in \Omega ^1(M)$, then

$$\displaystyle \begin{aligned} \varphi^*(d\omega)=d(\varphi^*\omega).\end{aligned}$$

Proof

The proof is easy if $\varphi$ is an orientation-preserving diffeomorphism: If $\hat {M}\subset \tilde {M}$ is any subdomain, then by Theorems 7.6, 7.11 and 7.15 we have

$$\displaystyle \begin{aligned} \int_{\hat{M}}\varphi^*(d\omega)= \int_{\varphi(\hat{M})} d\omega = \int_{\partial \varphi(\hat{M})} \omega = \int_{\partial \hat{M}} \varphi^*\omega \end{aligned} $$

By the uniqueness part of Theorem 7.15 we then must have $\varphi ^*(d\omega )=d(\varphi ^*\omega )$.

Unfortunately, here we only assume that $\varphi$ is a smooth map, so we have to rely on the coordinate formula provided in Theorem 7.15. We use the notation from the proof of Theorem 7.6 together with the equalities $a_{\tilde {v}}=b_{\tilde {u}}$ and $c_{\tilde {v}}=d_{\tilde {u}}$ that follow from the commutativity of partial derivatives of the component functions of $\varphi$ to compute:

$$\displaystyle \begin{aligned} & \quad \hspace{3.5pt} d(\varphi^*\omega)(\tilde{U},\tilde{V})\\ &= \omega(d\varphi(\tilde{V}))_{\tilde{u}}-\omega(d\varphi(\tilde{U}))_{\tilde{v}}\\ &= (b\cdot \omega(U)\circ \varphi +d\cdot \omega(V)\circ\varphi)_{\tilde{u}} - (a\cdot \omega(U)\circ \varphi +c\cdot \omega(V)\circ\varphi)_{\tilde{v}}\\ &= b(a\omega(U)_u \circ \varphi +c \omega(U)_v \circ \varphi)+d(a\cdot\omega(V)_u\circ \varphi + c\cdot\omega(V)_v\circ \varphi \\ &\quad -a(b\cdot\omega(U)_u\circ\varphi + d\cdot \omega(U)_v\circ \varphi) -c(b\cdot\omega(V)_u\circ\varphi + d\cdot \omega(V)_v\circ \varphi) \\ &=(ad-bc)(\omega(V)_u-\omega(U)_v)\circ \varphi \\ &= (ad-bc)\,d\omega(U,V)\circ\varphi\\ &=d\omega(a\cdot U\circ \varphi+c\cdot V\circ\varphi, b\cdot U\circ \varphi+d\cdot V\circ\varphi)\\ &=d\omega(d\varphi(\tilde{U}),d\varphi(\tilde{V}))\\ &= (\varphi^*d\omega)(\tilde{U},\tilde{V}), \end{aligned} $$

which proves the claim. □