6 Multiple integrals
Read Stewart Chapter 15 and Thomas Chapter 15
Notations:
Rectangle \(R= [a_1,b_1]\times \dots \times [a_n, b_n] \subseteq \mathbb{R}^n\).
6.1 Basic definition
Definition 6.1 Let \(f\) be a function on a rectangle \(R\). An n-fold Riemann sum for \(f\) over \(R\) is a sum of the following form \[\begin{equation*} \sum_{i_1=1}^{m_1}\dots \sum_{i_n=1}^{m_n} f(\xi_{i_1\dots i_n}) \Delta A \,, \end{equation*}\] where
\(\Delta A = \Delta x_1\times\dots \times \Delta x_n\),
\(\Delta x_i = (b_i-a_i)/m_i\),
\(\xi_{i_1\dots i_n}\in R_{i_1\dots i_n}\),
\(R_{i_1\dots i_n}= \prod [a_i + (i_1-1)\Delta x, a_i+ i_1\Delta x_i]\).
Definition 6.2 The double integral of \(f\) over a rectangle \(R \subseteq \mathbb{R}^2\) is \[\begin{equation*} \iint_{R} f(x,y) \, dA = \lim_{m,n\to \infty} \sum_{i=1}^m \sum_{j=1}^n f(\xi_{ij}) \Delta A \end{equation*}\] if the limit exists.
The triple integral of \(f\) over a rectangle \(R \subseteq \mathbb{R}^3\) is \[\begin{equation*} \iiint_{R} f(x,y) \, dA = \lim_{m,n,l\to \infty} \sum_{i=1}^m \sum_{j=1}^n \sum_{k=1}^l f(\xi_{ijk}) \Delta A \end{equation*}\] if the limit exists.
6.2 Iterated integrals
Suppose that \(f\) is integrable on \(R= [a,b]\times [c,d]\). An iterated integral of \(f\) is defined as \[\begin{equation*} \int_a^b A(x) \, dx \,, \end{equation*}\] where \[\begin{equation*} A(x) = \int_c^d f(x,y) \, dy \,. \end{equation*}\] Typically, we write the above as \[\begin{equation*} \int_a^b \int_c^d f(x,y) \, dy dx \,. \end{equation*}\] This means that we integrate in \(y\) before in \(x\)– always integrate the inner part first.
Similarly, we can define an iterated integral in a different order \[\begin{equation*} \int_c^d \int_a^b f(x,y) \, dx dy \,. \end{equation*}\]
The biggest question: Is it true that \[\begin{equation*} \int_a^b \int_c^d f(x,y) \, dy dx = \int_c^d \int_a^b f(x,y) \, dx dy \,? \end{equation*}\]
Theorem 6.1 (Special case of Fubini) If \(f\) is continuous on the rectangle \(R\), then \[\begin{equation*} \iint_R f(x,y) \, dA = \int_a^b \int_c^d f(x,y) \, dy dx = \int_c^d \int_a^b f(x,y) \, dx dy \,. \end{equation*}\]
Example 6.1 Let \[\begin{equation*} f(x,y) = \frac{x^2 - y^2}{(x^2 + y^2)^2} \,. \end{equation*}\] \[\begin{equation*} \int_0^1\int_0^1 f(x,y) \,dy dx = \frac{\pi}{4} = - \int_0^1\int_0^1 f(x,y) \, dx dy \,. \end{equation*}\]
Everything we discuss here is true for three-variable functions.
6.3 Change of coordinates
A coordinate transformation is a function \(\varphi\), which is bijective and differentiable for which \(D\varphi\) is invertible at all points in the domain. Here, \[\begin{equation*} D\varphi = \begin{pmatrix} \partial_1 \varphi_1 & \partial_2 \varphi_1 \\ \partial_1 \varphi_2 & \partial_2 \varphi_2 \end{pmatrix} \,. \end{equation*}\]
We will need to re-call the notion of invertible matrix here. For an \(n\times n\) matrix \(A\), it is invertible iff \(\det A \not= 0\),.
Theorem 6.2 Let \(f\) be a function of \((x,y)\) defined on the domain \(D\). Let \[\begin{equation*} \begin{pmatrix} x \\ y \end{pmatrix} = \varphi(u,v) \end{equation*}\] for some coordinate change function \(\varphi: D \to S\). If \(f\) is continuous and \(\varphi\) is differentiable, then \[\begin{equation*} \int_S f \, dA = \int_D f\circ \varphi |\det D \varphi| \, dA \end{equation*}\]
6.3.1 Applications of change of coordinates
6.3.1.1 Polar coordinate
In \(\mathbb{R}^2\), when the region of integration is a section of a disk centered at \(0\). Let \[\begin{equation*} \begin{pmatrix} x \\ y \end{pmatrix} = \varphi(r,\theta) = \begin{pmatrix} r\cos\theta \\ r\sin\theta \end{pmatrix} \,, \end{equation*}\] where \(a \leq r \leq b\) and \(\alpha \leq \theta \leq \beta\).
6.3.1.2 Cylindrical coordinate
In \(\mathbb{R}^3\), when the region of integration is part of a cylinder. Let \[\begin{equation*} \begin{pmatrix} x \\ y \\z \end{pmatrix} = \varphi(r,\phi,\theta) = \begin{pmatrix} r\cos\theta\\ r\sin\theta\\ z \end{pmatrix} \,, \end{equation*}\] where \(a \leq r \leq b\), \(\alpha \leq \theta \leq \beta\).
6.3.1.3 Spherical coordinate
In \(\mathbb{R}^3\), when the region of integration is a section of a ball centered at \(0\). Let \[\begin{equation*} \begin{pmatrix} x \\ y \\z \end{pmatrix} = \varphi(\rho,\phi,\theta) = \begin{pmatrix} \rho\sin\phi\cos\theta\\ \rho\sin\phi\sin\theta\\ \rho \cos\phi \end{pmatrix} \,, \end{equation*}\] where \(a \leq \rho \leq b\), \(\alpha \leq \theta \leq \beta\), and \(c \leq \phi \leq d\).