MA40254: Differential & Geometric Analysis

Contents

Motivation: the problem with grad, curl and div

Gradient

Let \(U\subseteq \R ^3\) be open and \(f\colon U\to \R \) differentiable. Then the partial derivatives of \(f\) define a vector field

\[ \grad f\colon U \to \R ^3;\quad x=\begin {pmatrix} x_1\\x_2\\x_3\end {pmatrix} \mapsto \begin {pmatrix} \partial f/\partial x_1\\ \partial f/\partial x_2\\ \partial f/\partial x_3\end {pmatrix} \]

i.e., \((\grad f)(x)\) is a vector at each \(x\in U\).

If \(\gamma \colon \R \to U;\;t\mapsto \gamma (t)\) is a curve with \(\gamma (0)=x\), we can ask if

\[ \frac {d\gamma }{dt} (0) = (\grad f)(x)\; ? \]

Now suppose we change to spherical polar coordinates by the map

\[ \varphi \colon (0,\infty )\times (0,\pi )\times (-\pi ,\pi )\to \R ^3;\quad \begin {pmatrix} r\\ \theta \\ \psi \end {pmatrix} \mapsto \begin {pmatrix} r \sin \theta \,\cos \psi \\ r \sin \theta \, \sin \psi \\ r\cos \theta \end {pmatrix} \]

If \(U\subseteq \im \varphi \) then \(x\), \(f\) and \(\gamma \) are represented in spherical polar coordinates by \(\tilde x=\varphi ^{-1}(x)\), \(\tilde f=f\circ \varphi \colon \varphi ^{-1}(U)\to \R \) and \(\tilde \gamma =\varphi ^{-1}\circ \gamma \colon \R \to \varphi ^{-1}(U)\).

Problem

To have

\[ \frac {d\gamma }{dt} (0) = (\grad f)(x)\; \Leftrightarrow \; \frac {d\tilde \gamma }{dt} (0) = (\grad \tilde f)(\tilde x) \]

we cannot define

\[ \grad \tilde f = \begin {pmatrix} \partial \tilde f/\partial r\\ \partial \tilde f/\partial \theta \\ \partial \tilde f/\partial \psi \end {pmatrix} \]

but instead must set

\[ \grad \tilde f = \begin {pmatrix} \partial \tilde f/\partial r\\ \frac {1}{r^2}\partial \tilde f/\partial \theta \\ \frac {1}{r^2(\sin \theta )^2}\partial \tilde f/\partial \psi \end {pmatrix}. \]

To understand this problem, recall that \(\grad f(x)\) is related to the (Fréchet) derivative \(Df_x\) of \(f\) at \(x\) by

\[ Df_x(v) =v\cdot (\grad f)(x) \]

where \(v\in \R ^3\) and \(\cdot \) denotes the Euclidean scalar/inner/dot product. Recall \(Df_x\colon \R ^3\to \R \) is the best linear approximation to \(f\) near \(x\), hence \(Df_x\in \R ^{3*}=\call (\R ^3,\R )\), the dual space of linear forms \(\R ^3\to \R \).

Root of problem

\(\grad f\) depends on the inner product, so only transforms nicely for diffeomorphisms which preserve the inner product, and \(\varphi \) above does not!

Solution

Work instead with

\[ D f = d f\colon U \to \R ^{3*};\quad x\mapsto D f_x\; ! \]

Divergence and curl

If \(v\colon U\to \R ^3\) is a vector field, we may define

\[ \mathrm {div}\, v= \frac {\partial v_1}{\partial x_1} + \frac {\partial v_2}{\partial x_2} + \frac {\partial v_3}{\partial x_3}\quad \text {and}\quad \mathrm {curl}\, v = \begin {pmatrix} \partial v_2/\partial x_3 - \partial v_3/\partial x_2\\ \partial v_3/\partial x_1 - \partial v_1/\partial x_3\\ \partial v_1/\partial x_2 - \partial v_2/\partial x_1 \end {pmatrix} \]

but the transformation rules into spherical polar coordinates are even more horrible.¹

We can resolve this problem by introducing differential forms: functions \(\alpha \) on \(U\) with values in \(\R =\Alt ^0(\R ^3)\), \(\R ^{3*}=\Alt ^1(\R ^3)\), \(\Alt ^2(\R ^3)\) and \(\Alt ^3(\R ^3)\), where \(\Alt ^k(\R ^3)\) denotes the vector space of alternating \(k\)-multilinear forms on \(\R ^3\). Then we replace \(\grad \), \(\mathrm {curl}\) and \(\mathrm {div}\) by the exterior derivative \(\d \) between functions with values in these spaces. This more sophisticated algebra simplifies the transformation law to

\[ \d \wt \alpha = \wt {\d \alpha }. \]

(Also \(\d ^2:=\d \circ \d =0\) captures in a memorable way the rules relating \(\grad \), \(\mathrm {curl}\) and \(\mathrm {div}\), and there is an obvious generalisation from \(\R ^3\) to \(\R ^n\).)

Integration

In vector calculus, integration is as important as differentiation, and there are line integrals, surface integrals and volume integrals: for example if \(x\colon U\to \R ^3\) parametrises a surface \(S\subseteq \R ^3\) (where \(U\subseteq \R ^2\)), and \(z\colon U\to \R ^3\) describes a vector field along the surface, then the surface integral of \(z\) along \(S\) is defined by

\[ \int _S z\cdot dS := \int _{(u,v)\in U} z(u,v)\cdot \Bigl (\frac {\partial x}{\partial u} \times \frac {\partial x}{\partial v}\Bigr ) du\,dv, \]

which again involves Euclidean geometry (not just the dot product, but the cross product).

Differential forms provide coordinate invariant reformulations of these definitions. In addition, the fundamental theorem of calculus, Stokes’ theorem for surfaces, and the divergence theorem for volumes are all special cases of Stokes’ theorem for differential forms \(\alpha \) on submanifolds \(M\) with boundary \(\partial M\):

\[ \int _M \d \alpha = \int _{\partial M} \alpha . \]

This includes in particular the Fundamental Theorem of Calculus when \(M=[a,b]\) is a closed interval:

\[ \int _{[a,b]} \frac {\d f}{\d x} \d x = \int _{\{a,b\}} f = f(b)-f(a). \]

1 Smooth functions on \(\R ^n\)

1.1 Differentiation

Recall that any normed vector space \(V\) is a metric space with \(d(x,y)=\norm {y-x}\). Hence any subset \(S\subseteq V\) is also a metric space, whose open and closed sets are intersections with \(S\) of open and closed subsets in \(V\) (respectively).

1.2 Inverse Function Theorem

For \(U,\wt U \subseteq \R ^n\) open and \(f : U \to \wt U\), \(g : \wt U \to U\) inverses, the differentiability of one does not imply the differentiability of the other. For example, for \(U=\wt U = \R \), \(x \mapsto x^3\) is differentiable but \(y \mapsto \sqrt [3]{y}\) is not (at \(y =0\)).

We often make use of the following corollary to the rank-nullity theorem: if \(\varphi \colon V\to W\) is a linear map between vector spaces of the same dimension and \(\ker \varphi =\{0\}\), then \(\varphi \) is a linear isomorphism.

Thus any diffeomorphism is necessarily a local diffeomorphism. This turns out to be sufficient, at least locally.

We will also see later that if \(f\) is \(C^k\) then \(f\colon U'\to \wt U\) is a \(C^k\) diffeomorphism.

To prove the Theorem 1.19, first note that if its conclusion holds for \(\tilde {f} = L \circ f\) for any linear isomorphism \(L\), it also holds for \(f\). If we take \(L := (Df_x)^{-1}\), then by the chain rule \(D\tilde f_u = DL_{f(u)} \circ Df_u = L \circ Df_u\) for any \(u\in U\), so in particular \(D\tilde {f}_x = \Id _{\R ^n}\). So without loss of generality, \(Df_x = \Id _{\R ^n}\).

Since \(f\) is \(C^1\), \(Df\) is continuous, so \(Df_u\) is close to \(\Id _{\R ^n}\) for \(u\) close to \(x\): concretely,

\begin{multline*} \qquad \exists \,r > 0 \quad \text {s.t.} \quad B_r(x) := \{u : \|u - x\| < r\} \subseteq U\\ \text {and}\quad \forall \, u\in B_r(x) \quad \text {we have}\quad \|\Id \nolimits _{\R ^n} - Df_u\|_{\mathrm {op}} < \half .\qquad \end{multline*} The plan is now to show that \(U' := B_r(x)\) satisfies the conclusions of Theorem 1.19. The key is to observe that if we define \(h := \Id _{\R ^n} - f\), then for \(y,z \in B_r(x)\), the mean value inequality implies that

\begin{equation} \label {eq:contract} \|h(y) - h(z) \| \leq \|y - z\| \sup _{u \in B_r(x)} \| \Id \nolimits _{\R ^n} - Df_u\|_{\text {op}} \leq \half \|y - z\| \\ \end{equation}

Now recall the following theorem from Analysis 2A.

Now (1.1) implies \(h\) is a contraction. Furthermore, for any \(w\in \R ^n\), the same is true for \(h_w\) defined by \(h_w(z)=h(z)+w = z - f(z) + w\), since \(h_w(y)-h_w(z)=h(y)-h(z)\). On the other hand, \(z\) is a fixed point of \(h_w\) (i.e., \(h_w(z)=z\)) if and only if \(f(z)=w\). We use this to prove that the image of \(U'\) is open.

Now for any \(v\in f(U')\) with \(U' = B_r(x)\subseteq U\), take \(y\in U'\) with \(f(y)=v\), and \(\delta >0\) such that \(B_\delta (y)\subseteq U'\). Then the above lemma applies to show \(f(U')\) contains an open ball centred at \(v=f(y)\). Thus \(f(U')\) is open, \(f\colon U'\to f(U')\) is a bijection between open sets, hence has an inverse \(g\colon f(U') \to U'\subseteq \R ^n\).

We now observe that any \(z\in U'\) could have been used in place of \(x\) in Lemma 1.23.

Hence \(g\) is differentiable on \(\wt U=f(U')\), which completes the proof of Theorem 1.19. In fact more is true: \(g\) is as differentiable as \(f\).

By Theorem 1.19, \(g\) is differentiable on \(\wt U\), hence \(C^0\). Theorem 1.25 now follows by induction on \(k\) using the following.

1.3 Implicit Function Theorem

For a differentiable function \(f : \R ^2 \to \R \), we can try to use the equation \(f(x,y) = 0\) to “implicitly” define \(y\) as a function of \(x\), i.e., find \(h : \R \to \R \) such that \(f(x,h(x)) = 0\) and the level set \(f^{-1}(0) = \{(x,y) \in \R ^2 \, | \, f(x,y) = 0\}\) is precisely \(\mathrm {graph}(h) = \{ (x,h(x)) \, | \, x \in \R \}\). The problem is that given a particular \(x\), there could be zero or multiple solutions \(y\) to the equation \(f(x,y) = 0\). For example, let \(f : \R ^2 \to \R \) be defined by \(f(x,y) = x - y^3 + 3y\).

Given \(x_0\) and \(y_0\) such that \(f(x_0,y_0) = 0\), we could try instead to define \(y=h(x)\) only for \(x\) close to \(x_0\), insisting that \(y\) is close to \(y_0\). However, this can still fail: in the example, if we take \((x_0, y_0) = (-2,1)\), then for \(x<-2\) there is no solution for \(y\), while for \(x>-2\) the solution is not unique. The problem here is that \(\frac {\partial f}{\partial y}=-3 y^2 + 3 = 0\). The Implicit Function Theorem asserts (in arbitrary dimensions) that this is the only problem.

To state it, we start with a function \(f\colon U\to \R ^m\) where \(U\) is open in \(\R ^{n+m}\cong \R ^n\times \R ^m\), and denote by \(D_1 f_z\) and \(D_2 f_z\) be the restrictions of \(Df_z\) to \(\R ^n\times \{0\}\cong \R ^n\) and \(\{0\}\times \R ^m\cong \R ^m\) (respectively) in \(\R ^n\times \R ^m\).

Henceforth, we use the term “diffeomorphism” to mean smooth (\(C^\infty \)) diffeomorphism, i.e., a smooth map with a smooth inverse. Using Definition 1.15, we may extend this terminology to functions \(f\colon S\to T\) between arbitrary subsets \(S \subseteq \R ^n\) and \(T \subseteq \R ^m\).

2 Submanifolds of \(\R ^s\)

2.1 Submanifolds and regular values

Unwinding Definition 1.15, this means that there is an open neighbourhood \(\wt U \subseteq \R ^s\) of \(x\) and smooth maps \(F : \wt U \to U'\) and \(\varphi : U' \to U\), with \(U=\wt U \cap M\), such that \(F|_{U} = \varphi ^{-1}\).

Finding parametrisations explicitly is usually rather tedious. Fortunately there is a more convenient general method for proving that a subset \(M\subseteq \R ^s\) is a submanifold using the implicit function theorem.

2.2 Tangent spaces and derivatives of maps between submanifolds

3 Differential forms

3.1 Motivation

Suppose \(U\sub \R ^n\) is open and \(\alpha \colon U\to \R ^{n*}=\calm ^1(\R ^n)\) is smooth. When does there exist a function \(f\colon U\to \R \) such that \(\alpha =Df\)?

Partial answer. If \(\alpha =Df\), then for all \(p\in U\), \((D\alpha _p)^\vee =D^2 f_p\in \calm ^2(\R ^n)\) is symmetric. Hence if we define, for any smooth \(\alpha \colon U\to \R ^{n*}\), any \(p\in U\) and any \(v_1,v_2\in \R ^n\), \(\d \alpha _p(v_1,v_2):= (D\alpha _p)^\vee (v_1,v_2)-(D\alpha _p)^\vee (v_2,v_1)\), then \(\alpha =Df\) implies \(\forall \,p\in U\), \(\d \alpha _p=0\) .

Because this is a natural question, we can expect it to behave well with respect to smooth changes of coordinates. Indeed if \(\varphi \colon U'\to U\) is a (local) diffeomorphism, and we define \(\varphi ^*f:=f\circ \varphi \colon U'\to \R \), then by the chain rule \(D(\varphi ^* f)_p = Df_{\varphi (p)}\circ D\varphi _p\). Hence if we define \((\varphi ^*\alpha )_p(v):=\alpha _{\varphi (p)}(D\varphi _p(v))\) then \(\alpha = Df\) if and only if \(\varphi ^*\alpha =D(\varphi ^*f)\).

A further computation with the chain rule shows that \(\d (\varphi ^*\alpha )_p= (\varphi ^* \d \alpha )_p\), where

\[ (\varphi ^* \d \alpha )_p(v_1,v_2) := d\alpha _{\varphi (p)}(D\varphi _p(v_1),D\varphi _p(v_2)). \]

Hence if \(\d \alpha _{\varphi (p)}=0\) then \(\d (\varphi ^*\alpha )_p=0\).

Notice that \(d\alpha _p(v,v)=0\), so that \(\d \alpha _p\in \calm ^2(\R ^n)\) is alternating. This calculus extends to differential forms, which are functions with values in the vector space \(\Alt ^k(\R ^n)\) of alternating \(k\)-linear forms on \(\R ^n\). In this chapter we define vector spaces \(\Omega ^k(U)\) of differential \(k\)-forms on \(U\), linear operators \(\d \colon \Omega ^k(U)\to \Omega ^{k+1}(U)\), called exterior derivatives, and, for any smooth \(\varphi \colon U'\to U\), linear operators \(\varphi ^*\colon \Omega ^k(U)\to \Omega ^k(U')\) called pullbacks such that:

• • \(\Omega ^0(U)\) is the space of smooth functions \(f\colon U\to \R \) and \(\d f = Df\in \Omega ^1(U)\);

• • For any \(\alpha \in \Omega ^k(U)\), \(\d (\d \alpha )=0\);

• • \(\d (\varphi ^*\alpha )=\varphi ^*\d \alpha \).

In addition, there is an associative multiplication on differential forms, and all of this structure can be extended from open subsets \(U\) of \(\R ^n\) to arbitrary submanifolds \(M\).

3.2 Alternating forms

Recall that if \(V\) is a real vector space then \(\calm ^k(V)=\calm ^k(V;\R )\) is the vector space of maps \(\alpha \colon V^k \to \R \), which are \(k\)-(multi)linear, i.e., for all \(i\) (etc.),

\[ \alpha (v_1,\dotsc ,v_{i-1},\lambda v_i + \mu w_i ,v_{i+1},\dotsc ,v_k) = \lambda \alpha (v_1,\dotsc ,v_k) + \mu \alpha (v_1,\dotsc ,w_i,\dotsc ,v_k). \]

Recall that for each \(k\in \N \), there is a symmetric group \(S_k\) of permutations \(\sigma \) of \(\{1,\dotsc ,k\}\), that any \(\sigma \in S_k\) is a composite of transpositions, and that the sign homomorphism \(\sgn : S_k \to \{\pm 1 \}\) is characterised by \(\sgn (\tau ) = -1\) for all transpositions \(\tau \); let \(A_k=\ker (\sgn )\).

Clearly \(\Id \cdot \alpha =\alpha \), and if \(\sigma ,\tau \in S_k\), consider \(w_i := v_{\sigma (i)}\); then \(w_{\tau (j)} = v_{\sigma (\tau (j))}\), so

\begin{align*} (\sigma \cdot (\tau \cdot \alpha ))(v_1,\dotsc ,v_k) &= (\tau \cdot \alpha )(w_1,\dotsc ,w_k) = \alpha (w_{\tau (1)},\dotsc ,w_{\tau (k)})\\ &= \alpha (v_{\sigma \circ \tau (1)}, \dotsc , v_{\sigma \circ \tau (k)}) = ((\sigma \circ \tau )\cdot \alpha )(v_1,\dotsc ,v_k). \end{align*} Hence \(\sigma \cdot (\tau \cdot \alpha ) = (\sigma \circ \tau )\cdot \alpha \), i.e., \(S_k \times \calm ^k(V) \to \calm ^k(V);\; (\sigma , \alpha ) \mapsto \sigma \cdot \alpha \) defines a (left) action of \(S_k\) on \(\calm ^k(V)\).

3.3 Differential forms and pullback

3.4 The exterior derivative on open subsets

For any \(\alpha \in \Omega ^k(U)\), the derivative of \(\alpha \) (which is smooth, by definition of \(\Omega ^k(U)\)) at \(p\in U\) in the usual sense is \((D\alpha )_p : \R ^n \to \Alt ^k(\R ^n)\). Using Notation 1.7, we then have \((D\alpha )_p^\vee \in \calm ^{k+1}(\R ^n)\) defined by

\[ (D\alpha )_p^\vee (v_1,v_2,\dotsc ,v_{k+1}) := (D\alpha )_p (v_1)(v_2,\dotsc ,v_{k+1}). \]

However, there is no reason to expect that \((D\alpha )_p^\vee \) is alternating (although it is alternating in the last \(k\) arguments).

In order to compute \(\d \alpha \) in coordinates, we need a bit more algebra.

The converse is false in general; however it does hold on \(\R ^n\) (if \(k > 0\)).

We will prove this later.

3.5 The wedge product and Leibniz rule

The wedge product of differential forms is bilinear and associative. In particular, if \(\alpha =f \d x_I\) and \(\beta =f \d x_J\) then \(\alpha \wedge \beta = fg\, \d x_I\wedge \d x_J\), hence also \(f\wedge \beta =f\beta \) and \(\alpha \wedge f = f \alpha \). Also \(\alpha \wedge \beta =(-1)^{k\ell }\beta \wedge \alpha \) and there is the following Leibniz/product rule.

3.6 Pullbacks and the exterior derivative on submanifolds

Let \(M\subseteq U\subseteq \R ^s\) with \(U\) open and \(M\) an \(m\)-dimensional submanifold. Then the inclusion map \(\iota =\Id _U|_M\colon M\to U\) is smooth, with derivative \(D\iota _p\colon T_p M\to \R ^s\) for any \(p\in M\). Motivated by pullback, for any \(\beta \in \Omega ^k(U)\) we would like to define a “differential form” \(\alpha =\iota ^*\beta \) on \(M\) by

\[ \alpha _p=(\iota ^*\beta )_p = (D\iota _p)^*(\beta _{\iota (p)})=\beta _p|_{T_pM^k} \in \Alt ^k(T_pM) \]

for any \(p\in M\). In order to provide \(\alpha \) with a fixed codomain, we let

\[ \mathcal {A}_m^k(\R ^s) := \bigsqcup _W \Alt ^k(W), \]

where the disjoint union is taken over all \(m\)-dimensional subspaces \(W \subseteq \R ^s\). This allows us to define differential forms on submanifolds as (local) pullbacks.

For any smooth map \(\varphi \colon N\to M\) between submanifolds \(M\) and \(N\) and any \(\alpha \in \Omega ^k(M)\), this lemma applies to \(N\cap \wt U\) and \(M\cap U\) for sufficiently small open neighbourhoods of any \(q\in N\) and \(\varphi (q)\in M\) such that \(\alpha =i^*\beta \) on \(U\cap M\) (\(i\colon U\cap M \to U\)) and \(\varphi \) has a smooth extension \(\widetilde \varphi \colon \wt U\to U\). Hence \(\varphi ^*\alpha \in \Omega ^k(N)\), i.e., is smooth.

Secondly, suppose that \(\varphi \colon \wt U\to U\cap M\) is a parametrisation of \(M\) (for \(\wt U\sub \R ^n\) open) and \(\alpha \in \Omega ^k(M)\) agrees with \(i^*\beta \) on \(U\cap M\) (\(i\colon U\cap M \to U\)). Then the lemma applies with \(\widetilde \varphi =\varphi \circ i\) and \(j=\id _{\wt U}\) to give \(\varphi ^* i^*\d \beta = \d (\varphi ^*\alpha )\) on \(\wt U\) and hence for any \(p\in U\cap M\),

\[ (i^*\d \beta )_p = ((\varphi ^{-1})^* \d (\varphi ^*\alpha ))_p \in \Alt ^{k+1}(T_p M). \]

Thus \((i^*\d \beta )_p\) depends only on \(\alpha \), not on the choice of local extension \(\beta \).

Now for any smooth map \(\varphi \colon N\to M\) between submanifolds \(M\) and \(N\) and any \(\alpha \in \Omega ^k(M)\), applying Lemma 3.39 on sufficiently small open neighbourhoods as above, we obtain that \(\varphi ^*\d \alpha =\d (\varphi ^*\alpha )\in \Omega ^k(N)\), generalizing Theorem 3.35.

3.7 Proof of the Poincaré Lemma

We turn the method of Example 3.26 into an algorithm. Note first that the two lists

\[ \epsilon _I : I \subseteq \{1,\dotsc ,n-1\}, |I| = k \quad \textrm {and} \quad \epsilon _n \wedge \epsilon _I : I \subseteq \{1,\dotsc ,n-1\}, |I| = k-1 . \]

combine to give a basis for \(\Alt ^k(\R ^n)\). Therefore, if for any \(k\) we let \(B^k_n \leq \Alt ^k(\R ^n)\) be the subspace spanned by the first list, then any \(\psi \in \Alt ^k(\R ^n)\) can be written uniquely as

\[ \psi = \nu + \epsilon _n \wedge \eta \]

for \(\nu \in B^k_n\) and \(\eta \in B^{k-1}_n\). Hence for any \(\alpha \in \Omega ^k(\R ^n)\), there exists a unique function \(\call (\alpha ) : \R ^n \to B^{k-1}_n\) such that for all \(p \in \R ^n\), \((\alpha - \d x_n \wedge \call (\alpha ))_p \in B^k_n\). We now observe that if \(\alpha \) doesn’t involve \(\d x_n\), then \(\d \alpha \) will be the sum of \(\d x_n \wedge \frac {\partial \alpha }{\partial x_n}\) and some terms that do not involve \(\d x_n\).

4 Integration and Stokes’ Theorem

4.1 Submanifolds with boundary

Let \(H^n\) be the closed half-space \(\{ p \in \R ^n \,|\, x_1(p) \leq 0 \}\), and let \(\partial H^n={\{0\} \times \R ^{n-1} \subset H^n}\). For \(U\) open in \(H^n\), let \(\partial U = U \cap \partial H^n\).

If \(f\colon H^n\to \R ^m\) is smooth, then \(Df_p\) is well-defined at all \(p\in H^n\), including \(p\in \partial H^n\), since \(D\tilde f_p\) is independent of the choice of smooth local extension \(\tilde f\colon \wt U\to \R ^m\) of \(f\) to an open neighbourhood \(\wt U\) of \(p\) in \(\R ^n\): observe that for \(v\in H^n\setminus \partial H^n\), \(\tilde f(p+tv) - \tilde f(p)= f(p+tv) - f(p)\) for \(t>0\), so \(D\tilde f_p(v)\) is determined by \(f\), and such \(v\) span \(\R ^n\).

4.2 Multiple integrals

We impose compact support to ensure convergence of the integrals in the following. For \(f \in C^0_c(H^n)\), we define \(\tilde {f}\in C^0_c(H^{n-1})\) as follows: for \(n\geq 2\) let

\[ \tilde {f} : H^{n-1} \to \R , \qquad p \mapsto \int _{\R } f(x_1(p),\dotsc ,x_{n-1}(p),t) \, \d t; \]

for \(n = 1\), \(H^1=(-\infty ,0]\), and \(\tilde {f}\in \R \) is the integral of \(f\) over this interval.

Given \(\varphi : \wt U \to U\) a diffeomorphism of open subsets of \(H^n\), define the Jacobian \(J_\varphi : \wt U \to \R \) by \(J_\varphi (p) = \det (D\varphi _p)\). Let \(f \in C^0_c(U)\) and note that \(\supp (f\circ \varphi ) = \varphi ^{-1}(\supp (f)) \subseteq \wt U\) is the continuous image of a compact set, and thus compact. Hence \(f\circ \varphi \) has compact support, so \((f\circ \varphi )\, |J_\varphi |\colon p\mapsto f(\varphi (p)) |J_\varphi (p)|\) is in \(C^0_c(\wt U)\)

A proof of this theorem is given in Appendix B, in the case of open subsets of \(\R ^n\) rather than \(H^n\), but the proof in the latter case is similar.

4.3 Integration of forms

Recall that \(\dim \Alt ^n(\R ^n) =1\) with basis \(\mathrm {Det}\), and that for any \(\alpha \in \Alt ^n(\R ^n)\) and linear map \(\phi \colon \R ^n\to \R ^n\), \(\phi ^*\alpha =\det (\phi )\alpha \) (exercise). Now suppose \(\alpha = f \, \d y_1 \wedge \cdots \wedge \d y_n\in \Omega ^n(U)\) where \(U\) is open in \(H^n\) with coordinates \(y_1,\ldots , y_n\) and \(f\colon U \to \R \) smooth. If \(\varphi \colon \wt U \to U\) is a diffeomorphism for \(\wt U\) open in \(H^n\) (with coordinates \(x_1,\ldots ,x_n\)), then

\begin{align*} (\varphi ^*\alpha )_p &=(\varphi ^*(f \,\d y_1 \wedge \cdots \wedge \d y_n))_p = (D\varphi _p)^*\bigl (f(\varphi (p))\mathrm {Det}\bigr ) = f(\varphi (p)) J_\varphi (p) \mathrm {Det}\\ &= f(\varphi (p))J_\varphi (p) (\d x_1\wedge \cdots \wedge \d x_n)_p. \end{align*} We say \(\varphi \) is orientation-preserving if \(\forall p \in \wt U\), \(J_\varphi (p)>0\); then \(J_\varphi (p) = |J_\varphi (p)|\), so the transformation rule for \(\varphi ^*\alpha \) resembles the change of variables formula of Theorem 4.6.

We write \(\alpha \in \Omega ^n_c(U)\) if \(\alpha = f \, \d y_1 \wedge \cdots \wedge \d y_n\in \Omega ^n(U)\) with \(f\in C^0_c(U)\).

4.4 Orientations

An orientation of an \(n\)-dimensional real vector space \(V\) is an element of the \(2\) element set \((\Alt ^n(V) \backslash \{0\}) /\sim \), where \(\alpha \sim \wt \alpha \) if \(\wt \alpha =\lambda \alpha \) for some \(\lambda \in \R ^+\).

The outward normal convention ensures that for \(U\sub H^n\) the standard orientation of \(U\) induces the standard orientation of \(\partial U\).

4.5 The integration map

For a SMWB \(M\) and \(\alpha \in \Omega ^k(M)\), we let \(\supp (\alpha )=\overline {\{p\in M:\alpha _p\neq 0\}}\sub M\), and write \(\alpha \in \Omega ^k_c(M)\) if \(\supp (\alpha )\) is compact.

To prove the existence and uniqueness of integration maps, we need a technical tool.

A proof is given in Appendix A.

4.6 Stokes’ theorem

Let \(i : \partial M \to M\) denote the inclusion of the boundary of an oriented SMWB \(M\); then any \(\alpha \in \Omega ^k(M)\) has a pullback \(i^*\alpha \in \Omega ^k(\partial M)\), and if \(\alpha \in \Omega ^k_c(M)\), then \(i^*\alpha \in \Omega ^k_c(\partial M)\). In particular, if \(\beta \in \Omega ^{n-1}_c(M)\), then \(\d \beta \in \Omega ^n_c(M)\) and \(i^*\beta \in \Omega ^{n-1}_c(\partial M)\) can be integrated on \(M\) and \(\partial M\) respectively.

To prove Stokes’ theorem, we may as well assume that \(\supp (\beta )\) is contained in \(U\) for some oriented parametrisation \(\varphi : \wt U \to U\), since any \(\beta \) can be written (using a partition of unity) as a sum of such forms. Then

\[ \int _M \d \beta = \int _{\wt U} \varphi ^*(\d \beta ) = \int _{\wt U} \d (\varphi ^*\beta ) \]

and (with \(i_{\wt U}\colon \partial \wt U \to \wt U\) being the inclusion)

\[ \int _{\partial M} i^*\beta = \int _{\partial \wt U} i^*_{\wt U}\varphi ^*\beta . \]

The theorem now follows by applying the next lemma to \(\gamma \in \Omega ^{n-1}_c(H^n)\) defined by

\[ \gamma _p = \begin {cases} (\varphi ^*\beta )_p & \textrm {for } p \in \wt U \\ 0 & \textrm {for } p \in H^n \setminus \supp (\varphi ^*\beta ). \end {cases} \]

A Existence of partitions of unity

Let \(U_i : i \in \mathcal {I}\) be an open cover of \(M\). For each \(U_i\), there is by definition an open subset \(\wt U_i \subseteq \R ^s\) such that \(U_i = M \cap \wt U_i\). Let \(\tilde M = \bigcup _{i \in \mathcal {I}} \wt U_i\) (an open subset of \(\R ^s\)). Then any partition of unity on \(\tilde M\) subordinate to \(\wt U_i : i \in \mathcal {I}\) induces a partition of unity on \(M\) subordinate to \(U_i : i \in \mathcal {I}\), so without loss, we can assume that \(M\) is open in \(\R ^s\).

Step 1: cover by a countable set of balls

Let \(\mathcal {V}\) be the set of subsets \(V \subseteq M\) such that:

• • there exist \(r, x_1,\ldots x_k\in \mathbb {Q}\) such that \(V=B_r(x)\) where \(x=(x_1,\ldots x_k)\);

• • the closure \(\overline V\) in \(\R ^s\) is contained in \(U_j\) for some \(j \in \mathcal {I}\).

\(\mathcal {V}\) is a countable set, so we may enumerate its elements as \(V_j:j\in \Z ^+\).

Set \(W_0=\varnothing \), \(\mathcal A_0=\mathcal V\) and, for \(m\in \Z ^+\),

\[ W_m=V_1\cup \cdots \cup V_m\quad \text {and}\quad \mathcal {A}_m = \{V\in \mathcal {V}: \overline V\cap W_m=\varnothing \}. \]

Then Claim 1, with \(W=W_m\), shows that \(\mathcal {A}_m\) covers \(M\setminus \overline {W_m}\): indeed \(\bigcup \mathcal {A}_m=M\setminus \overline {W_m}\).

Step 2: making the cover locally finite

We now define inductively for \(m\in \N \), a finite subset \(\mathcal {B}_m \subseteq \mathcal {V}\), such that \(\mathcal B_0=\varnothing \), and for \(m\in \Z ^+\), \(\mathcal B_m\) is covers \(\overline W_m\), so that \(\mathcal {A}_m \cup \mathcal {B}_m\) covers \(M\). To do this, observe that \(\overline {W_m}\subseteq M\) is a closed and bounded subset of \(\R ^s\), hence compact by Heine–Borel 4.3. Since (inductively) \(\mathcal {A}_{m-1} \cup \mathcal {B}_{m-1}\) covers \(M\), it has a finite subset \(\mathcal {B}_m\) which covers \(\overline {W_m}\).

We now set \(\mathcal {B} = \bigcup _{m \in \N } \mathcal {B}_m\), which is an open cover of \(\bigcup _{m\in \N } \overline {W_m}=M\). However, it is also “locally finite”: any \(p\in M\) belongs to \(W_m\) for some \(m\in \N \) and so if \(V\in \mathcal {B}\) with \(\overline V\cap W_m \neq \varnothing \), then \(V\notin \mathcal A_m\), and so \(V\in \mathcal B_1\cup \cdots \cup \mathcal B_{m-1}\), which is finite.

Step 3: defining the partition of unity

For each \(V = B_r(x) \in \mathcal {B}\), choose \(j(V)\in \mathcal {I}\) with \(\overline V \subseteq U_{j(V)}\), and define \(\rho _V : M \to \R \) by

\[ \rho _V(y) = \exp \left (-\frac {1}{r^2 - |x-y|^2}\right ) \]

for \(y \in B_r(x)\), and \(\rho _V(y) = 0\) for \(y \not \in B_r(x)\). Then \(\rho _V\) is smooth (exercise) with \(\supp (\rho _V) = \overline V\) and \(\rho _V\) positive on \(V\). Since \(\mathcal {B}\) is locally finite, each \(p \in M\) has an open neighbourhood \(W \subseteq M\) with \(\overline V \cap W \neq \emptyset \) for only finitely many \(V \in \mathcal {B}\). Hence the functions

\[ \sigma (x) = \sum _{V \in \mathcal {B}} \rho _V(x) \quad \text {and}\quad \sigma _i(x) = \sum _{V \in \mathcal {B}:j(V)=i} \rho _V(x) \]

(for \(i\in \mathcal I\)) are well-defined and smooth because only finitely many \(\rho _V\) are nonzero on an open neighbourhood of any point. Furthermore \(\sigma \) is nonvanishing, \(\supp (\sigma _i)\sub U_i\), and any point has an open neighbourhood which meets \(\supp (\sigma _i)\) for only finitely many \(i\in \mathcal I\). Hence \(\rho _i(x) := \sigma _i(x)/\sigma (x)\) defines a partition of unity subordinate to \(U_i:i\in \mathcal I\).

B Proof of the change of variables formula

For a diffeomorphism \(\varphi : V \to U\) of open subset of \(\R ^n\), let \(C(\varphi )\) be the statement:

\begin{equation} \label {eq:cov} \int _U f(y) \, \d y_1 \dotsb \, \d y_n =\int _V f(\varphi (x))|J_\varphi (x)| \, \d x_1 \dotsb \, \d x_n \end{equation}

for all \(f \in C^0_c(V)\). We wish to prove that \(C(\varphi )\) holds for any \(\varphi \). Main steps:

• 1. \(C(\varphi )\) and \(C(\psi ) \quad \Rightarrow \quad C(\varphi \circ \psi )\)

• 2. \(\varphi \) a permutation of coordinates\(\quad \Rightarrow \quad C(\varphi )\)

• 3. \(n = 1 \quad \Rightarrow \quad C(\varphi )\)

• 4. \(\varphi \) of the form \((x_1, \ldots , x_n) \mapsto (x_1, \ldots , x_{n-1}, h(x_1, \ldots , x_n))\quad \Rightarrow \quad C(\varphi )\).

• 5. for any \(\varphi \), any \(x \in V\) has an open neighbourhood \(V' \subseteq V\) such that \(\varphi |_{V'}\) is a composite of maps of the form (2) and (4).

• 6. using a partition of unity \(C(\varphi )\) holds for any \(\varphi \).

Step 1

Let \(U,V,W \subseteq \R ^n\) be open subsets, and let \(\varphi : V \to U\) and \(\psi : W \to V\) be diffeomorphisms. Then

\[ J_{\varphi \circ \psi }(x) = \det (D(\varphi \circ \psi )_x) = \det (D\varphi _{\psi (x)}\circ D\psi _x) = \det (D\varphi _{\psi (x)})\det (D\psi _x) = J_\varphi (\psi (x))J_\psi (x). \]

Now suppose \(C(\psi )\), i.e.,

\[ \int _W g(\psi (x))|J_\psi (x)| \, \d x_1 \dotsb \, \d x_n = \int _V g(y) \, \d y_1 \dotsb \, \d y_n \]

for any \(g \in C^0_c(V)\). Now if \(f \in C^0_c(U)\), then assuming \(C(\varphi )\) and applying \(C(\psi )\) with \(g = (f\circ \varphi )|J_\varphi | \in C^0_c(V)\), we obtain that

\begin{align*} \int _U f(z) \, \d z_1 \dotsb \, \d z_n &= \int _V f(\varphi (y))|J_\varphi (y)| \, \d y_1 \dotsb \, \d y_n\\ &= \int _W f(\varphi (\psi (x)))|J_\varphi (\psi (x))| |J_\psi (x)| \, \d x_1 \dotsb \, \d x_n \\ &=\int _W f((\varphi \circ \psi )(x))|J_{\varphi \circ \psi }(x)| \, \d x_1 \dotsb \, \d x_n, \end{align*} which gives \(C(\varphi \circ \psi )\).

Step 2

We want to show \(C(\varphi )\) when \(\exists \sigma \in S_n\) such that \(\varphi (x_1,\dotsc ,x_n) = (x_{\sigma (1)},\dotsc ,x_{\sigma (n)})\). Since this map is the restriction of a diffeomorphism \(s_\sigma : \R ^n \to \R ^n\), we can assume without loss that \(U = V = \R ^n\). Since \(|J_{s_\sigma }(x)| = 1\) for all \(x\), to establish \(C(s_\sigma )\), we need to show that we can change the order of the multiple integrals.

Let \(P \subseteq C^0_c(\R ^n)\) be the set of functions \(f : \R ^n \to \R \) such that \(f(x_1,\dotsc ,x_n) = f_1(x_1)\dotsb f_n(x_n)\) for some \(f_1,\dotsc ,f_n \in C^0_c(\R )\). Observe that for any such \(f\),

\begin{align*} \int _{\R ^n} f(x) \, \d x_1\dotsb \, \d x_n &= \int _{-\infty }^\infty f_1(t) \, \d t \,\dotsb \, \int _{-\infty }^\infty f_n(t) \,\d t = \int _{\R ^n} (f\circ s_\sigma ) \, \d x_1 \dotsb \, \d x_n, \end{align*} since each one-variable integral above is a real number, and multiplication is commutative. Hence (B.1) holds for all \(f\in P\). By linearity of integration, it follows that (B.1) holds for all \(f \in \mathrm {span}(P)\), the linear span of \(P\).

To establish \(C(s_\sigma )\), i.e., that (B.1) holds for all \(f \in C^0_c(\R ^n)\), we suppose \(\epsilon >0\) and apply the following special case of the Stone–Weierstrass theorem.

Theorem. The span of \(P\) is uniformly dense in \(C^0_c(\R ^n)\), i.e., for any \(f \in C^0_c(\R ^n)\) and \(\epsilon > 0\), there exists \(g \in \mathrm {span}(P)\) such that \(\forall \, x \in \R ^n\) we have \(| f(x) - g(x)| < \epsilon \).

Evidently this also implies that \(\forall \, x \in \R ^n\), \(| f(s_\sigma (x)) - g(s_\sigma (x))| < \epsilon \). Since \(g\in \mathrm {span(P)}\) we now have

\begin{align*} &\left | \int _{\R ^n} f(x) \, \d x_1 \dotsb \, \d x_n - \int _{\R ^n} f(s_\sigma (x)) \, \d x_1 \dotsb \,\d x_n\right | \\ &\quad =\left | \int _{\R ^n} (f(x)-g(x)) \, \d x_1 \dotsb \, \d x_n - \int _{\R ^n} (f(s_\sigma (x)) - g(s_\sigma (x))) \, \d x_1\dotsb \, \d x_n\right |\\ &\quad \leq \left | \int _{\R ^n} (f(x)-g(x)) \, \d x_1 \dotsb \, \d x_n\right | +\left |\int _{\R ^n} (f(s_\sigma (x)) - g(s_\sigma (x))) \, \d x_1\dotsb \, \d x_n\right |\leq 2\epsilon V, \end{align*} where \(V\) is the volume of a ball containing the supports of \(f,g,f\circ s_\sigma \) and \(g \circ s_\sigma \) (which exists as these functions all have compact support). Since \(\epsilon >0\) is arbitrary, the left-hand side is zero. In other words \(C(s_\sigma )\) holds.

Step 3

Suppose \(n=1\), so \(\varphi : V \to U\) for \(V,U \subseteq \R \) disjoint unions of open intervals, and \(J_\varphi (x) = \diff {}{\varphi }{x}\). Now given \(f \in C^0_c(U)\), then there exists a finite union of bounded open intervals \(V' \subseteq V\) such that \(\supp (f\circ \varphi ) \subseteq V'\). Therefore without loss of generality, \(V\) is a single bounded interval \((a,b) \subseteq \R \) and \(U=\varphi ((a,b))\). Then by the change of variables formula for functions of one variable

\[ \int _{U} f(y) \, \d y = \pm \int _{\varphi (a)}^{\varphi (b)} f(y) \, \d y =\pm \int _a^b f(\varphi (x))\diff {}{\varphi }{x}\,\d x, \]

where the sign is positive if \(\varphi (a)<\varphi (b)\) and negative if \(\varphi (b)<\varphi (a)\). However, this sign is also the sign of \(\diff {}{\varphi }{x}\) at all \(x\in (a,b)\), so the right hand side is

\[ \int _{V} f(\varphi (x)) \Bigl |\diff {}{\varphi }{x}\Bigr | \,\d x \]

as required.

Step 4

Suppose \(\varphi : V \to U\) has the form \((x_1,\dotsc ,x_n) \mapsto (x_1,\dotsc ,x_{n-1},h(x_1,\dotsc ,x_n))\) for some function \(h\). Fix \((x_1,\dotsc ,x_{n-1}) \in \R ^{n-1}\) and set

\[ U' = \{s \in \R \, | \, (x_1,\dotsc ,x_{n-1},s) \in U\}, \quad \text { and } \quad V' = \{t \in \R \, | \, (x_1,\dotsc ,x_{n-1},t) \in V\} \]

and \(\varphi ' : V' \to U', \; t \mapsto h(x_1,\dotsc ,x_{n-1},t)\). Then

\[ J_{\varphi '}(t) = \diff {}{\varphi '}{t} = \frac {\partial h}{\partial t} = J_\varphi (x_1,\dotsc ,x_{n-1},t) \]

and so for any \(f \in C^0_c(U)\), \(C(\varphi ')\) (Step 3) implies

\[ \tilde f(x_1,\ldots , x_{n-1}):=\int _{U'} f(x_1,\ldots ,x_{n-1},s) \, \d s =\int _{V'} g(x_1,\ldots ,x_{n-1},t) \, \d t \]

where \(g = (f\circ \varphi )|J_\varphi | \in C^0_c(V)\). However, extending both integrands by zero to \(\R \), the multiple integral of \(f\) over \(\R ^n\) is the multiple integral of \(\tilde f\) over \(\R ^{n-1}\), which therefore equals the multiple integral of \(g\) over \(\R ^n\), proving \(C(\varphi )\).

Step 5

A diffeomorphism \(\varphi : V \to U\) is called a \(k\)-graph if it is of the form

\[ \varphi (x) = \varphi (x_1,\dotsc ,x_n) = (x_1,\dotsc ,x_{n-k},\varphi _{n-k+1}(x),\dotsc ,\varphi _n(x)). \]

So

• • \(\varphi \) is a \(0\)-graph \(\Leftrightarrow \varphi = \Id \);

• • \(\varphi \) is a \(1\)-graph \(\Leftrightarrow \varphi \) is a diffeomorphism of the form (4);

• • any \(\varphi \) is an \(n\)-graph.

We are interested in the \(k = n\) case of the following claim.

Step 6

Let \(\varphi : V \to U\) be any diffeomorphism. The preceding steps show that any \(x \in V\) has a neighbourhood \(V' \subseteq V\) such that \(C(\varphi |_{V'})\). Equivalently, for any \(f \in C^0_{c}(U)\) such that \(\supp (f) \subseteq \im (\varphi (V'))\) we have

\[ \int _V (f\circ \varphi )(x)|J_\varphi (x)| \,\d x_1 \dotsb \, \d x_n = \int _U f(y) \, \d y_1 \dotsb \, \d y_n \]

Let \(U_i:i\in \mathcal I\) be the family of images of such \(V'\). Then \(U_i\) is an open cover of \(M\). Let \(\rho _i:i\in \mathcal {I}\) be a partition of unity subordinate this cover. Then any \(f \in C^0_c(U)\) can be written as \(f = \sum _i \rho _i f\). Since \(\supp (\rho _i f) \subseteq U_i\), then

\begin{align*} \int _U f(y) \, \d y_1 \dotsb \, \d y_n &= \sum _i \int _U (\rho _i f)(y) \, \d y_1 \dotsb \, \d y_n = \sum _i \int _V ((\rho _i f)\circ \varphi )(x)|J_\varphi (x)| \, \d x_1\dotsb \, \d x_n \\ &= \int _V (f\circ \varphi )(x)|J_\varphi (x)| \, \d x_1 \dotsb \, \d x_n. \end{align*}

This concludes the proof of the change of variables formula.