Multiple Integrals and Change of Variables

date2026-07-15document_iddoc_924acccc7940e421da668719f471f9a4description反復積分、Fubiniの定理、一般領域上の積分、Jacobianによる変数変換を体系的に導入する講義である。prerequisites一変数の定積分 / 多変数関数と偏微分 / 接平面・連鎖律・Jacobiantype講義content_typelecturestatusactive

mathmultivariable-calculusmultiple-integralsfubinichange-of-variableslecture

data/lecture/math/calculus/integral-definition-riemann-sums-and-signed-area.lecture.n.md data/lecture/math/multivariable-calculus/tangent-planes-chain-rule-and-jacobian.lecture.n.md

1Introduction

This lecture formulates the total amount of a quantity distributed over a planar or spatial region as a multiple integral. We first explain when a double integral over a rectangle can be evaluated as an iterated integral, and then extend the method to general regions and changes of variables.

Just as a single-variable definite integral is a limit of sums over small subintervals, a double integral is a limit of sums over small rectangles. This construction describes area, volume, mass, and other accumulated quantities within a common framework.

2Double Integrals over Rectangles

Let $R = [a, b] \times [c, d]$ and let $f : R \to R$ be bounded. Partition $R$ into small rectangles $R_{i j}$ and choose a point $(x_{{i j}^{*}}, y_{{i j}^{*}})$ in each $R_{i j}$ . The double Riemann sum is

\sum_{i, j} f (x_{{i j}^{*}}, y_{{i j}^{*}}) Δ A_{i j} .

If these sums converge to a limit $I$ , independently of the choice of sample points, as the maximum width of the partition tends to $0$ , then $f$ is Riemann integrable on $R$ , and we define

\iint_{R} f (x, y) d A = I .

In particular, every continuous function on a closed rectangle $R$ is Riemann integrable on $R$ .

3Iterated Integrals and Fubini's Theorem

In the iterated integral

\int_{c}^{d} (\int_{a}^{b} f (x, y) d x) d y,

one first holds $y$ fixed and integrates with respect to $x$ , and then integrates the resulting function of $y$ . The reversed order

\int_{a}^{b} (\int_{c}^{d} f (x, y) d y) d x

is defined analogously.

3.1Theorem: Fubini's Theorem for Continuous Functions

If $f$ is continuous on the closed rectangle $R = [a, b] \times [c, d]$ , then both iterated integrals exist and

\iint_{R} f (x, y) d A = \int_{c}^{d} \int_{a}^{b} f (x, y) d x d y = \int_{a}^{b} \int_{c}^{d} f (x, y) d y d x .

Thus, after verifying continuity, one may choose the more convenient order of integration.

The theorem can be proved by reorganizing the double Riemann sums as Riemann sums in one variable at a time and controlling the approximation error through uniform continuity. This lecture applies the theorem computationally. Its continuous-function form does not apply directly to an unbounded function or an unbounded region.

3.2Example 1: An Iterated Integral over a Rectangle

Let $R = [0, 2] \times [0, 1]$ and $f (x, y) = x y$ . Since $f$ is continuous on $R$ , Fubini's theorem gives

\begin{aligned} \iint_R xy\,dA &=\int_0^1\int_0^2xy\,dx\,dy\\ &=\int_0^1\left[\frac{x^2y}{2}\right]_0^2dy\\ &=\int_0^1 2y\,dy=1. \end{aligned}\begin{aligned} \iint_R xy\,dA &=\int_0^1\int_0^2xy\,dx\,dy\\ &=\int_0^1\left[\frac{x^2y}{2}\right]_0^2dy\\ &=\int_0^1 2y\,dy=1. \end{aligned}

4Double Integrals over General Regions

Let $α, β \in C ([c, d])$ with $α (y) [PARSE ERROR: Undefined("Command(\"le\")")] β (y)$ . Suppose a region can be described, with $x$ integrated first, as

D = {(x, y) ∣ c [PARSE ERROR: Undefined("Command(\"le\")")] y [PARSE ERROR: Undefined("Command(\"le\")")] d, α (y) [PARSE ERROR: Undefined("Command(\"le\")")] x [PARSE ERROR: Undefined("Command(\"le\")")] β (y)} .

If $f$ is continuous on $D$ , then

\iint_{D} f (x, y) d A = \int_{c}^{d} \int_{α (y)}^{β (y)} f (x, y) d x d y .

To reverse the order, one must redescribe the same region as

D = {(x, y) ∣ a [PARSE ERROR: Undefined("Command(\"le\")")] x [PARSE ERROR: Undefined("Command(\"le\")")] b, γ (x) [PARSE ERROR: Undefined("Command(\"le\")")] y [PARSE ERROR: Undefined("Command(\"le\")")] δ (x)} .

The integral signs and bounds cannot be exchanged mechanically.

4.1Example 2: A Triangular Region

Let $D = {(x, y) ∣ 0 [PARSE ERROR: Undefined("Command(\"le\")")] x [PARSE ERROR: Undefined("Command(\"le\")")] 1, 0 [PARSE ERROR: Undefined("Command(\"le\")")] y [PARSE ERROR: Undefined("Command(\"le\")")] x}$ , and suppose that $f$ is continuous on $D$ . The same region can be described by $0 [PARSE ERROR: Undefined("Command(\"le\")")] y [PARSE ERROR: Undefined("Command(\"le\")")] 1$ and $y [PARSE ERROR: Undefined("Command(\"le\")")] x [PARSE ERROR: Undefined("Command(\"le\")")] 1$ . Hence

\int_{0}^{1} \int_{0}^{x} f (x, y) d y d x = \int_{0}^{1} \int_{y}^{1} f (x, y) d x d y .

Reversing the order requires reconstructing the inequalities that define the region.

5Change of Variables and the Jacobian Determinant

Let $T : S \to D$ be given by

T (u, v) = (x (u, v), y (u, v)) .

Its Jacobian matrix and Jacobian determinant are

J_{T} (u, v) = (\begin{matrix} x_{u} & x_{v} \\ y_{u} & y_{v} \end{matrix}), \det J_{T} = x_{u} y_{v} - x_{v} y_{u} .

The quantity $[PARSE ERROR: Undefined("Command(\"lvert\")")] \det J_{T} [PARSE ERROR: Undefined("Command(\"rvert\")")]$ is the local area-scaling factor when a small rectangle in the $u v$ -plane is mapped to the $x y$ -plane.

Here the closure $\overline{S}$ is the set obtained by adjoining the boundary of $S$ to the region. In this lecture, a bounded Jordan region means a bounded region whose boundary has area zero. A homeomorphism is a continuous bijection with a continuous inverse. A $C^{1}$ diffeomorphism is a homeomorphism for which both the map and its inverse are of class $C^{1}$ .

5.1Theorem: Change of Variables for Double Integrals

Let $S$ and $D$ be bounded Jordan regions. Suppose $T : \overline{S} \to \overline{D}$ is a homeomorphism between their closures and its restriction to the interiors is a $C^{1}$ diffeomorphism. Assume also that $\det J_{T} \neq 0$ in the interior of $S$ and that $f$ is continuous on $\overline{D}$ . Then

\iint_{D} f (x, y) d x d y = \iint_{S} f (T (u, v)) [PARSE ERROR: Undefined("Command(\"lvert\")")] \det J_{T} (u, v) [PARSE ERROR: Undefined("Command(\"rvert\")")] d u d v .

The absolute value ensures that an orientation-reversing coordinate transformation does not produce negative area.

In applications such as polar coordinates, one-to-one behavior or the condition $\det J_{T} \neq 0$ may fail at a boundary point or along a boundary curve. A finite set of points or a piecewise smooth curve contributes no area. One therefore applies the formula on interior subregions that exclude such boundary pieces and takes a limit as those subregions approach the boundary.

6Polar Coordinates

For the polar coordinate transformation

x = r \cos θ, y = r \sin θ,

the Jacobian determinant is

\det (\begin{matrix} \cos θ & - r \sin θ \\ \sin θ & r \cos θ \end{matrix}) = r .

Since $r [PARSE ERROR: Undefined("Command(\"ge\")")] 0$ , we have $[PARSE ERROR: Undefined("Command(\"lvert\")")] \det J_{T} [PARSE ERROR: Undefined("Command(\"rvert\")")] = r$ , so the area element is

d A = r d r d θ .

6.1Example 3: Area of a Disk

For the disk $D$ of radius $1$ , the new bounds are $0 [PARSE ERROR: Undefined("Command(\"le\")")] r [PARSE ERROR: Undefined("Command(\"le\")")] 1$ and $0 [PARSE ERROR: Undefined("Command(\"le\")")] θ [PARSE ERROR: Undefined("Command(\"le\")")] 2 π$ . Therefore,

\iint_{D} 1 d A = \int_{0}^{2 π} \int_{0}^{1} r d r d θ = \int_{0}^{2 π} \frac{1}{2} d θ = π .

Omitting the Jacobian factor $r$ fails to preserve the area of the original region.

7Extension to Triple Integrals

The triple integral

∭_{E} f (x, y, z) d V

is likewise defined as a limit of Riemann sums over small rectangular boxes. The three-variable version of Fubini's theorem applies to continuous functions on closed rectangular boxes and permits evaluation by iterated integration. In a three-dimensional change of variables, the volume-scaling factor $[PARSE ERROR: Undefined("Command(\"lvert\")")] \det J_{T} [PARSE ERROR: Undefined("Command(\"rvert\")")]$ replaces the area-scaling factor.

8Computational Procedure

Describe the region explicitly by inequalities or a diagram.
Verify that the integrand and region satisfy the hypotheses of the theorem being used.
Decide whether to retain Cartesian coordinates or use a change of variables.
If variables are changed, compute both the transformed region and $[PARSE ERROR: Undefined("Command(\"lvert\")")] \det J_{T} [PARSE ERROR: Undefined("Command(\"rvert\")")]$ .
Check the sign, physical dimension, and consistency with a known area or volume.

9Cautions

When reversing the order of integration, redescribe the same region with inequalities appropriate to the new order.
Do not substitute the variables while omitting the absolute value of the Jacobian determinant.
Before using the change-of-variables theorem, determine where the map is one-to-one and where its Jacobian determinant is nonzero.
Do not extend the continuous-function form of Fubini's theorem unconditionally to unbounded functions or unbounded regions.

10Related Lectures

data/lecture/math/calculus/partial-derivatives-and-multiple-integrals.lecture.n.md data/lecture/math/multivariable-calculus/parametrized-curves-and-surfaces.lecture.n.md