Unit 6

Integration in many variables

Line, surface and volume integrals, and the theorems that connect them.

8 chapters written · 63 definitions · 47 theorems · builds on Unit 5

The story

Integration in many variables, as in MATH 376: along curves (6.1 and 6.2), over regions (6.3), and over surfaces (6.5). The theorems of Green, Stokes and Gauss (6.4 and 6.6) all say the same thing: the total change inside a region equals what happens on its boundary.

Chapter 6.7 shows that they are one theorem, written with differential forms, and 6.8 uses them to read Maxwell's equations and conservation laws.

Chapters

The theorem chain

Every theorem in the unit, in the order it is proved. Each one is used by the ones after it. The label says whether it is on the 375/376 exams (T), an Apostol extra (A), or beyond both (B).

6.1 Paths and line integrals

Proposition · T Line integrals are linear and additive $\int_C (\lambda F + \mu G) \cdot dr = \lambda\int_C F \cdot dr + \mu\int_C G \cdot dr$ , and if $C$ is $C_1$ followed by $C_2$ then $\int_C F \cdot dr = \int_{C_1} F \cdot dr + \int_{C_2} F \cdot dr$ .
Theorem · T Independence of parametrization

6.2 Gradient fields and potentials

Theorem · T Second fundamental theorem for line integrals If $F = \nabla\varphi$ on an open set $S$ with $\varphi$ of class $C^1$ , then for any piecewise smooth path $r: [a, b] \to S$ , $\int_r F \cdot dr = \varphi(r(b)) - \varphi(r(a))$ .
Theorem · T First fundamental theorem for line integrals If $S$ is open and connected, $F$ is continuous and its line integral is independent of the path, then $\varphi(x) = \int_p^x F \cdot dr$ (from a fixed base point $p$ ) is a $C^1$ function with $\nabla\varphi = F$ .
Theorem · T Three equivalent descriptions of a gradient field For a continuous field on an open connected set, (a) $F$ is a gradient, (b) its line integral is independent of the path, and (c) its line integral around every closed piecewise smooth path is $0$ , are equivalent.
Theorem · T Necessary condition for a gradient If $F = (f_1, \dots, f_n)$ is $C^1$ on an open set and is a gradient, then $\partial f_i/\partial x_j = \partial f_j/\partial x_i$ for all $i, j$ ; in the plane, $P_y = Q_x$ .
Theorem · T Equal mixed partials suffice on a convex set If $S$ is open and convex, $F = (P, Q)$ is $C^1$ on $S$ and $P_y = Q_x$ , then $F$ is a gradient field on $S$ ; in space the same holds when the curl is zero.
Theorem · A Equal mixed partials suffice on a simply connected set If $S$ is open and simply connected in the plane and $F = (P, Q)$ is $C^1$ with $P_y = Q_x$ , then $F$ is a gradient on $S$ .
Theorem · A The work-energy theorem For a particle of mass $m$ moving under a force field $F$ , the work done in a time interval equals the change in kinetic energy $\frac12mv^2$ .

6.3 Multiple integrals

Proposition · T The integral of a step function does not depend on the partition Two partitions that both make $s$ constant on every open piece give the same sum.
Theorem · T Linearity, additivity and monotonicity of step integrals For step functions $s, t$ on the same $Q$ : $\iint(c_1s + c_2t) = c_1\iint s + c_2\iint t$ ; $\iint_Q s = \iint_{Q_1}s + \iint_{Q_2}s$ if a line cuts $Q$ into $Q_1$ and $Q_2$ ; and $s \le t$ implies $\iint s \le \iint t$ .
Theorem · T Epsilon-sandwich criterion for the double integral
Theorem · T Continuous functions on a rectangle are integrable If $f$ is continuous on a closed rectangle $Q$ , then $f$ is integrable on $Q$ .
Lemma · T The graph of a continuous function has content zero If $\varphi: [a, b] \to \mathbb R$ is continuous, its graph $\{(x, \varphi(x))\}$ has content zero.
Theorem · T Bounded with discontinuities of content zero is integrable If $f$ is bounded on a rectangle $Q$ and the set of points where $f$ is not continuous has content zero, then $f$ is integrable on $Q$ .
Theorem · T Fubini's theorem for continuous functions on a rectangle If $f$ is continuous on $Q = [a, b] \times [c, d]$ , then $\iint_Qf = \int_c^d\!\int_a^bf\,dx\,dy = \int_a^b\!\int_c^df\,dy\,dx$ .
Theorem · T Integral over a type I region If $S$ is type I and $f$ is continuous on $S$ , then $f$ is integrable on $S$ and $\iint_Sf = \int_a^b\!\int_{\varphi_1(x)}^{\varphi_2(x)}f(x, y)\,dy\,dx$ .
Theorem · T Integrals commute with uniform limits on a rectangle If $f_n$ are integrable on $Q$ and $f_n \to f$ uniformly, then $f$ is integrable and $\iint f_n \to \iint f$ ; in fact $|\iint f_n - \iint f| \le \lVert f_n - f \rVert_\infty\,\text{area}(Q)$ .

6.4 Green's theorem and change of variables

Theorem · T Green's theorem If $C$ is a positively oriented piecewise smooth Jordan curve bounding $R$ , and $P$ and $Q$ are $C^1$ on an open set containing $R$ , then $\oint_C P\,dx + Q\,dy = \iint_R (Q_x - P_y)\,dA$ .
Theorem · T Area as a line integral For a positively oriented piecewise smooth Jordan curve $\alpha(t) = (X(t), Y(t))$ bounding $S$ , and any real $c$ , $\text{area}(S) = \int_a^b\big[(c - 1)YX' + cXY'\big]\,dt$ ; with $c = 1, 0, \frac12$ : $\text{area}(S) = \oint x\,dy = -\oint y\,dx = \frac12\oint(x\,dy - y\,dx)$ .
Theorem · A Green's theorem for regions with holes If $C_1$ is a Jordan curve and $C_2, \dots, C_n$ are disjoint Jordan curves inside it, all counterclockwise, and $R$ is the region inside $C_1$ and outside the others, then $\iint_R(Q_x - P_y)\,dA = \oint_{C_1} - \oint_{C_2} - \dots - \oint_{C_n}$ of $P\,dx + Q\,dy$ , for $P, Q$ of class $C^1$ on an open set containing $R$ .
Corollary · A Invariance of a line integral under deformation If $P_y = Q_x$ on an open connected set $S$ and $C_2$ lies inside $C_1$ with the region between them in $S$ , then $\oint_{C_1}P\,dx + Q\,dy = \oint_{C_2}P\,dx + Q\,dy$ , both counterclockwise.
Theorem · T Change of variables in a double integral If $g: T \to S$ is one-to-one and $C^1$ with $J \ne 0$ (allowed to fail on a set of content zero) and $f$ is continuous on $S$ , then $\iint_Sf\,dx\,dy = \iint_Tf(X(u, v), Y(u, v))\,|J(u, v)|\,du\,dv$ .
Theorem · A Area under a change of variables

6.5 Surfaces, area and flux

Theorem · T Length and direction of the cross product $a \times b$ is orthogonal to $a$ and to $b$ ; $\lVert a \times b\rVert^2 = \lVert a\rVert^2\lVert b\rVert^2 - (a\cdot b)^2$ ; so $\lVert a\times b\rVert = \lVert a\rVert\lVert b\rVert\sin\theta$ is the area of the parallelogram with sides $a$ and $b$ . Also $b \times a = -a \times b$ and $e_1\times e_2 = e_3$ .
Theorem · T Normal and area element of a parametrized surface $N = r_u \times r_v$ is perpendicular to the tangent plane at each regular point, and the area element is $dS = \lVert r_u\times r_v\rVert\,du\,dv$ .
Proposition · T Normal of a graph For the graph $z = g(x, y)$ with $r(x, y) = (x, y, g(x, y))$ : $r_x \times r_y = (-g_x, -g_y, 1)$ , which points upward, and $\lVert r_x\times r_y\rVert = \sqrt{1 + g_x^2 + g_y^2}$ .
Theorem · T Flux as a double integral If $r_u\times r_v$ points the same way as $n$ , then $\iint_S F\cdot n\,dS = \iint_TF(r(u, v))\cdot(r_u\times r_v)\,du\,dv$ ; if it points the opposite way, the right side has the opposite sign.
Proposition · T Surface integrals do not depend on the parametrization

6.6 Curl, divergence, Stokes' theorem and Gauss's theorem

Theorem · T Identities for curl and divergence Curl and divergence are linear; $\operatorname{div}(\operatorname{curl}F) = 0$ for $C^2$ fields; $\operatorname{curl}(\nabla f) = 0$ for $C^2$ functions; $\operatorname{div}(fF) = f\operatorname{div}F + \nabla f\cdot F$ and $\operatorname{curl}(fF) = f\operatorname{curl}F + \nabla f\times F$ .
Theorem · T Stokes' theorem If $S$ is a piecewise smooth oriented surface whose boundary curve $\partial S$ carries the induced orientation and $F$ is $C^1$ on an open set containing $S$ , then $\oint_{\partial S}F\cdot dr = \iint_S(\operatorname{curl}F)\cdot n\,dS$ .
Proposition · B Curl is circulation per unit area For a $C^1$ field $F$ , a point $p$ and a unit vector $n$ , the circulation of $F$ around the circle of radius $\varepsilon$ about $p$ in the plane perpendicular to $n$ (counterclockwise from the tip of $n$ ), divided by its area $\pi\varepsilon^2$ , tends to $(\operatorname{curl}F)(p)\cdot n$ as $\varepsilon\to0$ .
Theorem · T The divergence theorem If $E$ is a solid region with boundary a closed piecewise smooth surface $S = \partial E$ with outward unit normal $n$ , and $F$ is $C^1$ on an open set containing $E$ , then $\iint_SF\cdot n\,dS = \iiint_E\operatorname{div}F\,dV$ .
Proposition · B Divergence is flux per unit volume For a $C^1$ field $F$ and a point $p$ , the outward flux of $F$ through the sphere $S_\varepsilon$ of radius $\varepsilon$ about $p$ , divided by the volume of the ball, tends to $(\operatorname{div}F)(p)$ as $\varepsilon\to0$ .
Corollary · A Green's theorem in normal form For a positively oriented Jordan curve $C$ bounding $D$ and a $C^1$ planar field $F = (P, Q)$ , $\oint_CF\cdot n\,ds = \iint_D(P_x + Q_y)\,dA$ , where $n$ is the outward unit normal and $ds$ is arc length.

6.7 Differential forms and the one big theorem

Theorem · B d is gradient, curl and divergence In $\mathbb R^3$ : $df = f_x\,dx + f_y\,dy + f_z\,dz$ (gradient); $d(P\,dx + Q\,dy + R\,dz) = (R_y - Q_z)\,dy\wedge dz + (P_z - R_x)\,dz\wedge dx + (Q_x - P_y)\,dx\wedge dy$ (curl); $d(A\,dy\wedge dz + B\,dz\wedge dx + C\,dx\wedge dy) = (A_x + B_y + C_z)\,dx\wedge dy\wedge dz$ (divergence).
Theorem · B d squared is zero For every $C^2$ form $\omega$ , $d(d\omega) = 0$ . For $\omega = f$ a function in $\mathbb R^3$ this is $\operatorname{curl}\nabla f = 0$ , and for a 1-form it is $\operatorname{div}\operatorname{curl}F = 0$ .
Theorem · B Poincare lemma On a star-shaped region (such as a ball or all of $\mathbb R^n$ ) every closed form of degree $\ge1$ is exact.
Proposition · B Forms contain the integrals of 6.1 and 6.5 For a curve, $r^*(P\,dx + Q\,dy + R\,dz) = F(r)\cdot r'\,dt$ , so $\int_C\omega$ is the line integral. For a surface, $r^*(A\,dy\wedge dz + B\,dz\wedge dx + C\,dx\wedge dy) = G(r)\cdot(r_u\times r_v)\,du\wedge dv$ , so $\iint_S\omega$ is the flux.
Proposition · B Pullback commutes with d For a $C^2$ map $\varphi$ and a form $\omega$ , $\varphi^*(d\omega) = d(\varphi^*\omega)$ .
Theorem · B Generalized Stokes theorem If $M$ is a compact oriented $k$ -dimensional region with boundary $\partial M$ carrying the induced orientation and $\omega$ is a $C^1$ $(k-1)$ -form on a neighbourhood of $M$ , then $\displaystyle\int_{\partial M}\omega = \int_Md\omega$ .

6.8 Maxwell's equations, conservation laws and heat

Lemma · B A continuous function with zero integral over every region is zero If $h$ is continuous on an open set and $\iiint_Eh\,dV = 0$ for every ball $E$ inside it, then $h = 0$ everywhere.
Theorem · B Gauss's law follows from Coulomb's law For charges at points $p_i$ with values $q_i$ and a closed surface $S$ not passing through any of them, $\iint_SE\cdot n\,dS = \frac1{\varepsilon_0}\sum_{p_i\text{ inside}}q_i$ .
Corollary · B The differential form of Gauss's law If Gauss's law holds for every closed surface, and $E$ is $C^1$ and $\rho$ continuous, then $\nabla\cdot E = \rho/\varepsilon_0$ at every point.
Theorem · B The continuity equation If the amount of a quantity in every fixed region $E$ changes only through the flux across $\partial E$ , then $\partial_tu + \nabla\cdot j = 0$ .
Theorem · B Electromagnetic waves In empty space ( $\rho = 0$ , $J = 0$ ) each Cartesian component of $E$ satisfies the wave equation $\partial_t^2E = c^2\nabla^2E$ with $c = 1/\sqrt{\mu_0\varepsilon_0}\approx2.998\times10^8$ m/s.
Theorem · B The heat equation from conservation of energy Energy conservation (continuity) plus Fourier's law gives $\rho c\,\partial_tT = \kappa\nabla^2T$ .

Terms this unit defines

Piecewise smooth path Line integral Work Reparametrization Closed path Circulation Line integral with respect to arc length Centre of mass Moment of inertia Gradient field Potential Path independence Convex set Simply connected set Potential energy Exact differential equation Step function on a rectangle Double integral Content zero Iterated integral Type I region Type II region Triple integral Jordan curve Positive orientation Winding number Cross product Parametric surface Fundamental vector product Surface area Surface integral Oriented surface Orientable surface Flux Curl Divergence Vorticity Irrotational field Solenoidal field Laplacian Induced orientation Solid region Closed surface One-form Two-form Differential form Wedge product Exterior derivative Closed form Exact form Pullback Electric field Charge density Gauss's law Magnetic field Current density Ampere's law Continuity equation Displacement current Faraday's law Maxwell's equations Incompressible flow Heat flux

Unit review

A mixed set of 12 quick checks drawn at random from the 77 in this unit's chapters. Each one says which chapter it comes from, so a miss tells you what to reread. Press "New set" for another mix.

Where it leads

Units that build on this one: Unit 9, Complex analysis, Unit 13, Geometry and topology.