Integration of Differential Forms

Let $H$ be a compact convex set in $\mathbb {R}^{k}$, with nonempty interior. Let $f \in \mathscr {C}(H)$, put $f(\mathbf{x}) = 0$ in the complement of $H$, and define $\int_{H} f$ as in Definition 10.3.

Prove that $\int_{H} f$ is independent of the order in which the $k$ integrations are carried out.

Hint: Approximate $f$ by functions that are continuous on $\mathbb {R}^{k}$ and whose supports are in $H$, as was done in Example 10.4.

For $i = 1,2,3, \ldots$, let $\varphi_{i} \in \mathscr {C}\left(\mathbb {R}^{1}\right)$ have support in $\left(2^{-i}, 2^{1-i}\right)$, such that $\int \varphi_{i} = 1$. Put

Then $f$ has compact support in $\mathbb {R}^{2}, f$ is continuous except at $(0,0)$, and

Observe that $f$ is unbounded in every neighborhood of $(0,0)$.

For $(x, y) \in \mathbb {R}^{2}$, define

Prove that $\mathbf{F} = \mathbf{G}_{2} \circ \mathbf{G}_{1}$, where

are primitive in some neighborhood of $(0,0)$. Compute the Jacobians of $\mathbf{G}_{1}, \mathbf{G}_{\mathbf{2}}, \mathbf{F}$ at $(0,0)$. Define

and find

so that $F = H_{1} \circ H_{2}$ is some neighborhood of $(0,0)$.

Formulate and prove an analogue of Theorem 10.8, in which $K$ is a compact subset of an arbitrary metric space. (Replace the functions $\varphi_{i}$ that occur in the proof of Theorem 10.8 by functions of the type constructed in Exercise 22 of Chap. 4).

Strengthen the conclusion of Theorem 10.8 by showing that the functions $\psi_{i}$ can be made differentiable, and even infinitely differentiable. (Use Exercise 1 of Chap. 8 in the construction of the auxiliary functions $\varphi_{i}$).

Let $H$ be the parallelogram in $\mathbb {R}^{2}$ whose vertices are $(1,1),(3,2),(4,5),(2,4)$. Find the affine map $T$ which sends $(0,0)$ to $(1,1),(1,0)$ to $(3,2),(0,1)$ to $(2,4)$. Show that $J_{T} = 5$. Use $T$ to convert the integral

to an integral over $I^{2}$ and thus compute $\alpha$.

Define $(x, y) = T(r, \theta )$ on the rectangle

by the equations

Show that $T$ maps this rectangle onto the closed disc $D$ with center at $(0,0)$ and radius $a$, that $T$ is one-to-one in the interior of the rectangle, and that $J_{T}(r, \theta ) = r$. If $f \in \mathscr {C}(D)$, prove the formula for integration in polar coordinates:

Hint: Let $D_{0}$ be the interior of $D$, minus the interval from $(0,0)$ to $(0, a)$. As it stands, Theorem 10.9 applies to continuous functions $f$ whose support lies in $D_{0}$. To remove this restriction, proceed as in Example 10.4.

Let $a \rightarrow \infty$ in Exercise 9 and prove that

for continuous functions $f$ that decrease sufficiently rapidly as $\left|x\right|+\left|y\right| \rightarrow \infty$. (Find a more precise formulation). Apply this to

to derive formula (101) of Chap. 8.

Define $(u, v) = T(s, t)$ on the strip

by setting $u = s-s t, v = s t$. Show that $T$ is a $1-1$ mapping of the strip onto the positive quadrant $Q$ in $\mathbb {R}^{2}$. Show that $J_{T}(s, t) = s$.

For $x > 0, y > 0$, integrate

over $Q$, use Theorem 10.9 to convert the integral to one over the strip, and derive formula (96) of Chap. 8 in this way. (For this application, Theorem 10.9 has to be extended so as to cover certain improper integrals. Provide this extension).

Let $I^{k}$ be the set of all $\mathbf{u} = \left(u_{1}, \ldots , u_{k}\right) \in \mathbb {R}^{k}$ with $0 \leq u_{i} \leq 1$ for all $i$; let $Q^{k}$ be the set of all $\mathrm{x} = \left(x_{1}, \ldots , x_{k}\right) \in \mathbb {R}^{k}$ with $x_{l} \geq 0, \Sigma x_{l} \leq 1$. ($I^{k}$ is the unit cube; $Q^{k}$ is the standard simplex in $\mathbb {R}^{k}$). Define $\mathbf{x} = T(\mathbf{u})$ by

Show that

Show that $T$ maps $I^{k}$ onto $Q^{k}$, that $T$ is $1-1$ in the interior of $I^{k}$, and that its inverse $S$ is defined in the interior of $Q^{k}$ by $u_{1} = x_{1}$ and

for $i = 2, \ldots , k$. Show that

and

Let $r_{1}, \ldots , r_{k}$ be nonnegative integers, and prove that

Hint: Use Exercise 12, Theorems 10.9 and 8.20. Note that the special case $r_{1} = \cdots = r_{k} = 0$ shows that the volume of $Q^{k}$ is $1 / k!$.

Prove formula (46).

Formula (46) of Section 10.14: every $k$-tuple $\left\{ j_{1}, \ldots , j_{k}\right\}$ of distinct integers $1 \leq j_{i} \leq n$ can be converted to an increasing $k$-index $J$ by a finite number of interchanges of pairs, each of which amounts to a multiplication by $-1$; hence

where $\varepsilon \left(j_{1}, \ldots , j_{k}\right)$ is $1$ or $-1$, depending on the number of interchanges that are needed. Formula (46) asserts that

where $s$ is as in Definition 9.33: for an ordered $k$-tuple of integers, $s\left(j_{1}, \ldots , j_{k}\right) = \prod_{p<q} \operatorname {sgn}\left(j_{q}-j_{p}\right)$, with $\operatorname {sgn} x = 1$ if $x > 0$, $\operatorname {sgn} x = -1$ if $x < 0$, $\operatorname {sgn} x = 0$ if $x = 0$. (Here, for an increasing $k$-index $I = \left\{ i_{1}, \ldots , i_{k}\right\}$, $1 \leq i_{1} < \cdots < i_{k} \leq n$, one writes $d x_{I} = d x_{i_{1}} \wedge \cdots \wedge d x_{i_{k}}$.)

If $\omega$ and $\lambda$ are $k$ - and $m$-forms, respectively, prove that

If $k \geq 2$ and $\sigma = \left[p_{0}, p_{1}, \ldots , p_{k}\right]$ is an oriented affine $k$-simplex, prove that $\partial^{2} \sigma = 0$, directly from the definition of the boundary operator $\partial$. Deduce from this that $\partial^{2} \Psi = 0$ for every chain $\Psi$.

Hint: For orientation, do it first for $k = 2, k = 3$. In general, if $i < j$, let $\sigma_{i j}$ be the $(k-2)$-simplex obtained by deleting $\mathbf{p}_{i}$ and $\mathbf{p}_{j}$ from $\sigma$. Show that each $\sigma_{i j}$ occurs twice in $\partial^{2} \sigma$, with opposite sign.

Put $J^{2} = \tau_{1}+\tau_{2}$, where

Explain why it is reasonable to call $J^{2}$ the positively oriented unit square in $\mathbb {R}^{2}$. Show that $\partial J^{2}$ is the sum of 4 oriented affine 1 -simplexes. Find these. What is $\partial \left(\tau_{1}-\tau_{2}\right)$?

Consider the oriented affine 3 -simplex

in $\mathbb {R}^{3}$. Show that $\sigma_{1}$ (regarded as a linear transformation) has determinant 1. Thus $\sigma_{1}$ is positively oriented.

Let $\sigma_{2}, \ldots , \sigma_{6}$ be five other oriented 3 -simplexes, obtained as follows: There are five permutations $\left(i_{1}, i_{2}, i_{3}\right)$ of $(1,2,3)$, distinct from $(1,2,3)$. Associate with each $\left(i_{1}, i_{2}, i_{3}\right)$ the simplex

where $s$ is the sign that occurs in the definition of the determinant. (This is how $\tau_{2}$ was obtained from $\tau_{1}$ in Exercise 17).

Show that $\sigma_{2}, \ldots , \sigma_{6}$ are positively oriented. Put $J^{3} = \sigma_{1}+\cdots +\sigma_{6}$. Then $J^{3}$ may be called the positively oriented unit cube in $\mathbb {R}^{3}$.

Show that $\partial J^{3}$ is the sum of 12 oriented affine 2 -simplexes. (These 12 triangles cover the surface of the unit cube $I^{3}$).

Show that $\mathrm{x} = \left(x_{1}, x_{2}, x_{3}\right)$ is in the range of $\sigma_{1}$ if and only if $0 \leq x_{3} \leq x_{2} \leq x_{1} \leq 1$.

Show that the ranges of $\sigma_{1}, \ldots , \sigma_{6}$ have disjoint interiors, and that their union covers $I^{3}$. (Compare with Exercise 13; note that $3! = 6$).

Let $J^{2}$ and $J^{3}$ be as in Exercise 17 and 18. Define

These are affine, and map $\mathbb {R}^{2}$ into $\mathbb {R}^{3}$. Put $\beta_{r i} = B_{r i}\left(J^{2}\right)$, for $r = 0,1, i = 1,2,3$. Each $\beta_{r i}$ is an affine-oriented 2-chain. (See Sec. 10.30). Verify that

in agreement with Exercise 18.

State conditions under which the formula

is valid, and show that it generalizes the formula for integration by parts. Hint: $d(f \omega ) = (d f) \wedge \omega +f d \omega$.

As in Example 10.36, consider the 1 -form

in $\mathbb {R}^{2}-\left\{ 0\right\}$.

As in Example 10.37, define $\zeta$ in $\mathbb {R}^{3}-\left\{ 0\right\}$ by

where $r = \left(x^{2}+y^{2}+z^{2}\right)^{1 / 2}$, let $D$ be the rectangle given by $0 \leq u \leq \pi , 0 \leq v \leq 2 \pi$, and let $\Sigma$ be the 2 -surface in $\mathbb {R}^{3}$, with parameter domain $D$, given by

Fix $n$. Define $r_{k} = \left(x_{1}^{2}+\cdots +x_{k}^{2}\right)^{1 / 2}$ for $1 \leq k \leq n$, let $E_{k}$ be the set of all $\mathbf{x} \in \mathbb {R}^{n}$ at which $r_{k} > 0$, and let $\omega_{k}$ be the $(k-1)$-form defined in $E_{k}$ by

Note that $\omega_{2} = \eta , \omega_{3} = \zeta$, in the terminology of Exercises 21 and 22. Note also that

Let $\omega = \Sigma a_{i}(\mathbf{x}) d x_{i}$ be a 1 -form of class $\mathscr {C}^{\prime \prime }$ in a convex open set $E \subset \mathbb {R}^{n}$. Assume $d \omega = 0$ and prove that $\omega$ is exact in $E$, by completing the following outline:

Fix $\mathbf{p} \in E$. Define

Apply Stokes' theorem to affine-oriented 2-simplexes [p, x, y] in E. Deduce that

for $\mathbf{x} \in E, \mathbf{y} \in E$. Hence $\left(D_{i} f\right)(\mathbf{x}) = a_{i}(\mathbf{x})$.

Assume that $\omega$ is a 1-form in an open set $E \subset \mathbb {R}^{n}$ such that

for every closed curve $\gamma$ in $E$, of class $\mathscr {C}^{\prime }$. Prove that $\omega$ is exact in $E$, by imitating part of the argument sketched in Exercise 24.

Assume $\omega$ is a 1 -form in $\mathbb {R}^{3}-\left\{ 0\right\}$, of class $\mathscr {C}^{\prime }$ and $d \omega = 0$. Prove that $\omega$ is exact in $\mathbb {R}^{3}-\left\{ 0\right\}$.

Hint: Every closed continuously differentiable curve in $\mathbb {R}^{3}-\left\{ 0\right\}$ is the boundary of a 2-surface in $\mathbb {R}^{3}-\left\{ 0\right\}$. Apply Stokes' theorem and Exercise 25.

Let $E$ be an open 3-cell in $\mathbb {R}^{3}$, with edges parallel to the coordinate axes. Suppose $(a, b, c) \in E, f_{i} \in \mathscr {C}^{\prime }(E)$ for $i = 1,2,3$,

and assume that $d \omega = 0$ in $E$. Define

where

for $(x, y, z) \in E$. Prove that $d \lambda = \omega$ in $E$. Evaluate these integrals when $\omega = \zeta$ and thus find the form $\lambda$ that occurs in part $(e)$ of Exercise 22.

Fix $b > a > 0$, define

for $a \leq r \leq b, 0 \leq \theta \leq 2 \pi$. (The range of $\Phi$ is an annulus in $\mathbb {R}^{2}$). Put $\omega = x^{3} d y$, and compute both

to verify that they are equal.

Prove the existence of a function $\alpha$ with the properties needed in the proof of Theorem 10.38, and prove that the resulting function $F$ is of class $\mathscr {C}^{\prime }$. (Both assertions become trivial if $E$ is an open cell or an open ball, since $\alpha$ can then be taken to be a constant. Refer to Theorem 9.42).

If $\mathbf{N}$ is the vector given by (135), prove that

Also, verify Eq. (137).

Let $E \subset \mathbb {R}^{3}$ be open, suppose $g \in \mathscr {C}^{\prime \prime }(E), h \in \mathscr {C}^{\prime \prime }(E)$, and consider the vector field

Fix $\delta , 0 < \delta < 1$. Let $D$ be the set of all $(\theta , t) \in \mathbb {R}^{2}$ such that $0 \leq \theta \leq \pi ,-\delta \leq t \leq \delta$. Let $\Phi$ be the 2 -surface in $\mathbb {R}^{3}$, with parameter domain $D$, given by

where $(x, y, z) = \Phi (\theta , t)$. Note that $\Phi (\pi , t) = \Phi (0,-t)$, and that $\Phi$ is one-to-one on the rest of $D$.

The range $M = \Phi (D)$ of $\Phi$ is known as a Möbius band. It is the simplest example of a nonorientable surface.

Prove the various assertions made in the following description: Put $\mathbf{p}_{1} = (0,-\delta ), \mathbf{p}_{2} = (\pi ,-\delta ), \mathbf{p}_{3} = (\pi , \delta ), \mathbf{p}_{4} = (0, \delta ), \mathbf{p}_{5} = \mathbf{p}_{1}$. Put $\gamma_{i} = \left[\mathbf{p}_{i}, \mathbf{p}_{i+1}\right]$, $i = 1, \ldots , 4$, and put $\Gamma_{i} = \Phi \circ \gamma_{i}$. Then

Put $\mathbf{a} = (1,0,-\delta ), \mathbf{b} = (1,0, \delta )$. Then

and $\partial \Phi$ can be described as follows. $\Gamma_{1}$ spirals up from a to $\mathbf{b}$; its projection into the $(x, y)$-plane has winding number +1 around the origin. (See Exercise 23, Chap. 8). $\Gamma_{2} = [b, a]$. $\Gamma_{3}$ spirals up from a to $\mathbf{b}$; its projection into the $(x, y)$ plane has winding number -1 around the origin. $\Gamma_{4} = [b, a]$. Thus $\partial \Phi = \Gamma_{1}+\Gamma_{3}+2 \Gamma_{2}$. If we go from $\mathbf{a}$ to $b$ along $\Gamma_{1}$ and continue along the "edge" of $M$ until we return to a, the curve traced out is