Multiple Integrals

The integration theory of several variables (analysis spine). We extend the Riemann integral (riemann-integral.md) to functions on boxes and Jordan-measurable regions of $R^{n}$ , prove Fubini's theorem (iterated integration) and the change-of-variables formula with its Jacobian determinant. These are the tools that compute the volume, flux, and circulation integrals appearing in the integral theorems.

References: Rudin, Principles of Mathematical Analysis, ch. 10; Spivak, Calculus on Manifolds, ch. 3.

1. Integration over Boxes

A box (rectangle) is $R = [a_{1}, b_{1}] \times \dots \times [a_{n}, b_{n}]$ ; it plays the role of the domain of integration, the $R^{n}$ analogue of the interval $[a, b]$ . The integrand is a bounded function $f : R \to R$ defined on that box — so $R$ is where $f$ lives and $f$ supplies a value $f (x)$ at each point $x \in R$ . A partition $P$ of $R$ is a choice, for each edge $i$ , of a one-dimensional partition

$a_{i} = t_{i, 0} < t_{i, 1} < \dots < t_{i, k_{i}} = b_{i} .$

These grids together cut $R$ into closed subboxes: for any index tuple $(j_{1}, \dots, j_{n})$ with $1 \leq j_{i} \leq k_{i}$ ,

$Q = i = 1 \prod n [t_{i, j_{i} - 1}, t_{i, j_{i}}] .$

The volume (or $n$ -dimensional content) of such a subbox is the product of its edge lengths,

$vol (Q) = i = 1 \prod n (t_{i, j_{i}} - t_{i, j_{i} - 1}) \geq 0,$

and this is likewise the definition of $vol (R) = \prod_{i = 1}^{n} (b_{i} - a_{i})$ for the box itself. The subboxes have pairwise disjoint interiors, cover $R$ , and their volumes are additive: $\sum_{Q} vol (Q) = vol (R)$ (each factor $b_{i} - a_{i} = \sum_{j_{i}} (t_{i, j_{i}} - t_{i, j_{i} - 1})$ telescopes, and the product of sums expands to the sum over tuples). For bounded $f : R \to R$ , define lower and upper Darboux sums exactly as in one variable,

$L (f, P) = Q \sum (Q in f f) vol (Q), U (f, P) = Q \sum (Q sup f) vol (Q),$

the sum running over all subboxes $Q$ of $P$ . $f$ is integrable if $sup_{P} L = in f_{P} U$ , the common value being

$\int_{R} f = \int_{R} f (x) d x^{1} \dots d x^{n} .$

The Riemann criterion ( $U (f, P) - L (f, P) < ε$ ) and the proof that continuous functions on a box are integrable (via uniform continuity on the compact box, metric-spaces.md §5) carry over verbatim from riemann-integral.md.

2. Jordan Measure and General Regions

To integrate over non-box regions $E$ , extend $f$ by $0$ outside $E$ and integrate $f 1_{E}$ over a big box. This works when the boundary is negligible.

A set has (Jordan) measure zero if it can be covered by finitely many boxes of arbitrarily small total volume. (Lebesgue measure zero allows countably many — the distinction the integrability criterion below needs.)
$E$ is Jordan measurable if $\partial E$ has measure zero; then its volume is $vol (E) = \int 1_{E}$ .

Lebesgue's criterion (stated). A bounded $f$ on a box is Riemann integrable iff its discontinuity set has (Lebesgue) measure zero.

Smooth hypersurfaces (graphs of $C^{1}$ functions, regular level sets from inverse-implicit.md) have measure zero, so regions bounded by them — disks, balls, the solid regions in the divergence theorem — are Jordan measurable and continuous integrands over them are integrable.

3. Fubini's Theorem

Fubini's Theorem. If $f$ is continuous (or bounded and integrable with a measure-zero discontinuity set) on $R = A \times B$ with $A \subseteq> R^{p}$ , $B \subseteq R^{q}$ boxes, then $\int_{R} f = \int_{A} (\int_{B} f (x, y) d y) d x = \int_{B} (\int_{A} f (x, y) d x) d y .$

Proof (Darboux sums — click to expand).

Assume $f$ continuous (so every inner integral below exists; the general case is handled in the remark that follows). Write points of $R$ as $(x, y)$ with $x \in A$ , $y \in B$ , and set the inner integral

$g (x) = \int_{B} f (x, y) d y .$

We show $g$ is integrable on $A$ with $\int_{A} g = \int_{R} f$ ; the other order is identical by symmetry.

Step 1 — pair up partitions. Let $P_{A}$ partition $A$ into subboxes $S$ and $P_{B}$ partition $B$ into subboxes $T$ . Their product $P = P_{A} \times P_{B}$ partitions $R$ into subboxes $S \times T$ with $vol (S \times T) = vol (S) vol (T)$ , so

$L (f, P) = S \sum T \sum (S \times T in f f) vol (T) vol (S) .$

Step 2 — bound the inner sum by $g$ . Fix a subbox $S$ and any point $x \in S$ . Since ${x} \times T \subseteq S \times T$ , we have $in f_{S \times T} f \leq in f_{y \in T} f (x, y)$ , hence

$T \sum (S \times T in f f) vol (T) \leq T \sum (y \in T in f f (x, y)) vol (T) = L (f (x, \cdot), P_{B}) \leq \int_{B} f (x, y) d y = g (x) .$

This holds for every $x \in S$ , so the left-hand side (independent of $x$ ) is a lower bound for $g$ on $S$ , giving $\sum_{T} (in f_{S \times T} f) vol (T) \leq in f_{S} g$ .

Step 3 — squeeze. Multiplying by $vol (S)$ and summing over $S$ ,

$L (f, P) \leq S \sum (S in f g) vol (S) = L (g, P_{A}) .$

The mirror argument with suprema gives $U (g, P_{A}) \leq U (f, P)$ . Combining with the trivial $L (g, P_{A}) \leq U (g, P_{A})$ ,

$L (f, P) \leq L (g, P_{A}) \leq U (g, P_{A}) \leq U (f, P) .$

Because $f$ is integrable, $sup_{P} L (f, P) = in f_{P} U (f, P) = \int_{R} f$ ; the outer two terms squeeze together, forcing $L (g, P_{A})$ and $U (g, P_{A})$ to the same limit. Hence $g$ is integrable on $A$ and $\int_{A} g = \int_{R} f$ . $□$

The general (integrable) case. If $f$ is merely bounded and integrable, the inner integral $\int_{B} f (x, \cdot)$ need not exist for every $x$ . Replace $g$ by the lower and upper inner integrals $\underline{g} (x) = \underline{\int_{B}} > f (x, \cdot)$ and $\overline{g} (x) = \overline{\int_{B}} f (x, \cdot)$ . Steps 1–3 give $L (f, P) \leq L (\underline{g}, P_{A}) \leq U (\overline{g}, P_{A}) \leq U (f, P)$ , so both $\underline{g}$ and $\overline{g}$ are integrable on $A$ with integral $\int_{R} f$ ; since $\underline{g} \leq \overline{g}$ and their integrals agree, $\underline{g} => \overline{g}$ except on a measure-zero set, and the iterated integral equals $\int_{R} f$ . (This is why the hypothesis is stated as "integrable," not just "each slice integrable.")

A multiple integral is computed as iterated one-variable integrals, and the

order may be swapped. For a region described as $E = {(x, y) : x \in [a, b], φ (x) \leq y \leq ψ (x)}$ ("type I"), this gives $\int_{E} f = \int_{a}^{b} \int_{φ (x)}^{ψ (x)} f (x, y) d y d x$ . Fubini is the practical engine for every multiple integral and is invoked repeatedly in the proofs of Green's and the divergence theorem (integral-theorems.md).

Hypotheses matter. Without integrability/absolute-integrability the iterated integrals can differ. The classic counterexample $\int_{0}^{1} \int_{0}^{1} > \frac{x ^{2} - y ^{2}}{( x ^{2} + y ^{2} ) ^{2}} d y d x = \frac{π}{4} \neq = - \frac{π}{4} =$ (other order) shows Fubini fails for non-integrable integrands.

4. Change of Variables

Change-of-Variables Theorem. Let $g : U \to V$ be a $C^{1}$ diffeomorphism between open sets in $R^{n}$ (bijective, with $C^{1}$ inverse), and let $f$ be integrable on $V$ . Then $\int_{V} f (y) d y = \int_{U} f (g (x)) det D g (x) d x .$

The factor $∣ det D g ∣$ — the Jacobian determinant — is the local volume scaling of the map: $D g$ sends an infinitesimal box of volume $d x$ to a parallelepiped of volume $∣ det D g ∣ d x$ (determinant = oriented volume of the image of the unit cube). This is the $n$ -dimensional generalization of the substitution rule's $φ^{'}$ factor (riemann-integral.md §6), with $∣ \cdot ∣$ appearing because unoriented volume is intrinsically positive.

Proof idea. Locally $g$ is approximated by the linear map $D g (a)$ ; for linear maps the determinant volume-scaling is elementary, and a partition-of-unity / covering argument with the inverse function theorem (inverse-implicit.md) globalizes it. $□$

Standard cases.

Polar $(x, y) = (r cos θ, r sin θ)$ : $∣ det D g ∣ = r$ , so $d x d y = r d r d θ$ .
Spherical $(r, θ, ϕ)$ : $∣ det D g ∣ = r^{2} sin θ$ , so $d V = r^{2} sin θ d r d θ d ϕ$ .
Cylindrical $(r, θ, z)$ : $∣ det D g ∣ = r$ .

5. Improper and Parameter-Dependent Integrals

For unbounded regions or integrands, define multiple integrals as limits over exhausting compact sets; for nonnegative integrands the limit is independent of the exhaustion. Differentiation under the integral sign (Leibniz rule): if $f$ and $\partial f / \partial t$ are continuous, then $\frac{d}{d t} \int_{A} f (x, t) d x = \int_{A} \frac{\partial f}{\partial t} (x, t) d x$ . These reductions all ultimately rest on the uniform-convergence theorems of sequences-series.md.

6. Orientation: A Preview

The change-of-variables formula carries an absolute value $∣ det D g ∣$ because it measures unoriented volume. The integral theorems, by contrast, are sensitive to orientation (which way a boundary is traversed, which way a normal points), and there the signed determinant $det D g$ is what appears. Reconciling these — unsigned volume vs signed orientation — is precisely the role of differential forms (differential-forms.md), where integration is defined to transform by $det D g$ (not its absolute value) and orientation is built in from the start.

7. Where this page is used

Fubini is the computational core of the proofs in integral-theorems.md.
Jacobian determinant / change of variables defines surface area and flux integrals (vector-calculus.md) and reappears as the transformation law for $n$ -forms (differential-forms.md).
Jordan-measurable regions are the domains for the divergence theorem (integral-theorems.md).

Next: Vector Calculus — the classical integral operators and integrals.

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