Multiple Integrals
5.1 The Concept of Multiple Integrals
A subset of \(\mathbb{R}^m\) of the form \[ \begin{aligned} R &= \{(x^1, x^2, \dots, x^m) \in \mathbb{R}^m \mid a^k \le x^k \le b^k, k=1, 2, \dots, m\} \\ &= [a^1, b^1] \times [a^2, b^2] \times \dots \times [a^m, b^m] \end{aligned} \] is called an \(m\)-dimensional rectangle in \(\mathbb{R}^m\), denoted as \[ \begin{aligned} \mu_m(R) := (b^1 - a^1)(b^2 - a^2) \dots (b^m - a^m) \end{aligned} \] which is called the volume of \(R\). When it does not lead to confusion, we also denote \(\mu(R) = \mu_m(R)\).
A collection \(\{R_k\}_{k=1}^p\) is called a partition \(\mathcal{P}\) of the rectangle \(R\) if each \(R_k\) is an \(m\)-dimensional rectangle in \(\mathbb{R}^m\), and \[ \begin{aligned} \bigcup_{k=1}^p R_k = R, \quad \text{int} R_i \cap \text{int} R_j = \emptyset \quad (\text{for any } 1 \le i < j \le p). \end{aligned} \]
Given a function \(f: R \to \mathbb{R}\), for a partition \(\mathcal{P} = \{R_k\}_{k=1}^p\) of the rectangle \(R\) and a corresponding set of tagged points \(\xi = \{\xi_k\}_{k=1}^p\) (where each \(\xi_k \in R_k\)), let \[ \begin{aligned} S(f, \mathcal{P}, \xi) = \sum_{k=1}^p f(\xi_k) \mu(R_k) \end{aligned} \] This is called the Riemann sum of \(f\) on \(R\) with respect to the partition \(\mathcal{P}\) and the set of tagged points \(\xi\).
We say that \(f\) is Riemann integrable on \(R\) if there exists \(I \in \mathbb{R}\) such that for any \(\epsilon > 0\), there exists \(\delta > 0\) such that for any partition \(\mathcal{P}\) of \(R\), as long as \[ \begin{aligned} \|\mathcal{P}\| = \max_{1 \le k \le p} \left( \sup_{x,y \in R_k} \|x - y\| \right) < \delta \end{aligned} \] we have \(|S(f, \mathcal{P}, \xi) - I| < \epsilon\) for any set of tagged points \(\xi\). At this time, we denote \[ \begin{aligned} I = \int_R f(x) d\mu = \int_R f(x) dx = \int_R f(x) dx_1 \dots dx_m, \end{aligned} \] and call it the Riemann integral of \(f\) on \(R\).
Let \(f: R \to \mathbb{R}\) be a bounded function on a rectangle \(R\), and \(\mathcal{P} = \{R_k\}_{k=1}^p\) be a partition of \(R\). Let \[ \begin{aligned} \overline{S}(f, \mathcal{P}) = \sum_{k=1}^p \sup_{x \in R_k} f(x) \mu(R_k), \quad \underline{S}(f, \mathcal{P}) = \sum_{k=1}^p \inf_{x \in R_k} f(x) \mu(R_k), \end{aligned} \] which are called the Darboux upper sum and Darboux lower sum of \(f\) on \(R\) with respect to the partition \(\mathcal{P}\), respectively.
A partition \(\mathcal{Q} = \{\tilde{R}_j\}_{j=1}^q\) of \(R\) is called a refinement of \(\mathcal{P} = \{R_k\}_{k=1}^p\) if each rectangle \(R_k\) in \(\mathcal{P}\) is a union of some rectangles in \(\mathcal{Q}\).
It is easy to see that for any two partitions \(\mathcal{P}_1, \mathcal{P}_2\) of the rectangle \(R\), there exists a common refinement denoted by \(\mathcal{P}_1 \lor \mathcal{P}_2\), and \[ \begin{aligned} \underline{S}(f, \mathcal{P}_1) \le \underline{S}(f, \mathcal{P}_1 \lor \mathcal{P}_2) \le \overline{S}(f, \mathcal{P}_1 \lor \mathcal{P}_2) \le \overline{S}(f, \mathcal{P}_2). \end{aligned} \]
Let \(\mathfrak{P}\) be the set of all possible partitions of the rectangle \(R\). We call \[ \begin{aligned} \overline{S}(f) = \inf_{\mathcal{P} \in \mathfrak{P}} \overline{S}(f, \mathcal{P}), \quad \underline{S}(f) = \sup_{\mathcal{P} \in \mathfrak{P}} \underline{S}(f, \mathcal{P}) \end{aligned} \] the Darboux upper integral and Darboux lower integral of \(f\) on \(R\), respectively.
If \(\overline{S}(f) = \underline{S}(f)\), then \(f\) is said to be Darboux integrable on \(R\). Their common value is denoted as \[ \begin{aligned} I = \int_R f(x) d\mu = \int_R f(x) dx = \int_R f(x) dx_1 \dots dx_m \end{aligned} \] and is called the Darboux integral of \(f\) on \(R\).
Theorem 5.1 The following conclusions are equivalent:
- \(f\) is Riemann integrable on the rectangle \(R\);
- \(f\) is bounded and Darboux integrable on the rectangle \(R\);
- \(f\) is bounded on the rectangle \(R\), and the set of discontinuities of \(f\) in \(R\) is a set of volume zero (measure zero);
- For any \(\epsilon > 0\), there exists a partition \(\mathcal{P}\) of the rectangle \(R\) such that \[ \begin{aligned} \overline{S}(f, \mathcal{P}) - \underline{S}(f, \mathcal{P}) < \epsilon. \end{aligned} \]
If \(f\) is Riemann (Darboux) integrable on \(R\), then the Riemann integral of \(f\) is equal to the Darboux integral of \(f\).
Theorem 5.2 Let \(R \subset \mathbb{R}^m\) be an \(m\)-dimensional rectangle. Let \(\mathscr{R}(R)\) be the set of all Riemann integrable functions on \(R\), and \(\mathscr{C}(R)\) be the set of all continuous functions on \(R\). Then:
(Linearity) \(\mathscr{R}(R)\) is a linear space, and \(\mathscr{C}(R)\) is a linear subspace of \(\mathscr{R}(R)\). Furthermore, for any \(f, g \in \mathscr{R}(R)\) and any \(\alpha, \beta \in \mathbb{R}\): \[ \begin{aligned} \int_R (\alpha f(x) + \beta g(x)) d\mu = \alpha \int_R f(x) d\mu + \beta \int_R g(x) d\mu; \end{aligned} \]
(Monotonicity) If \(f, g \in \mathscr{R}(R)\) satisfy \(f(x) \le g(x)\) for all \(x \in R\), then: \[ \begin{aligned} \int_R f(x) d\mu \le \int_R g(x) d\mu; \end{aligned} \]
\(f \in \mathscr{R}(R)\) if and only if \(f_+, f_- \in \mathscr{R}(R)\), where: \[ \begin{aligned} f_+(x) &= \max\{f(x), 0\} = \frac{|f(x)| + f(x)}{2}, \\ f_-(x) &= \max\{-f(x), 0\} = \frac{|f(x)| - f(x)}{2}. \end{aligned} \] Thus \(f = f_+ - f_-\) and \(|f| = f_+ + f_-\). Furthermore: \[ \begin{aligned} \left| \int_R f(x) d\mu \right| \le \int_R |f(x)| d\mu; \end{aligned} \]
(Cauchy-Schwarz Inequality) For any \(f, g \in \mathscr{R}(R)\), it follows that \(fg \in \mathscr{R}(R)\), and: \[ \begin{aligned} \left| \int_R f(x)g(x) d\mu \right| \le \sqrt{\int_R |f(x)|^2 d\mu \cdot \int_R |g(x)|^2 d\mu}; \end{aligned} \]
(Mean Value Theorem for Integrals) If \(f \in \mathscr{C}(R)\), \(g \in \mathscr{R}(R)\), and \(g(x) \ge 0\) for all \(x \in R\), then there exists \(\eta \in R\) such that:
\[ \begin{aligned} \int_R f(x)g(x) d\mu = f(\eta) \int_R g(x) d\mu. \end{aligned} \]
5.2 Jordan Measurable Sets and Riemann Integrals
Let \(D \subset \mathbb{R}^m\) be a non-empty bounded subset. For a function \(f : D \to \mathbb{R}\), we choose a rectangle \(R \subset \mathbb{R}^m\) that covers \(D\) (i.e., \(D \subseteq R\)), and define \[ \begin{aligned} \tilde{f}(x) = \begin{cases} f(x), & x \in D; \\ 0, & x \in R \setminus D, \end{cases} \end{aligned} \] If \(\tilde{f}\) is Riemann integrable on \(R\), then we say \(f\) is Riemann integrable on \(D\), and denote \[ \begin{aligned} \int_D f(x) d\mu = \int_R \tilde{f}(x) d\mu. \end{aligned} \]
A set \(D \subset \mathbb{R}^m\) is called a Jordan measurable set if the boundary set \(\partial D\) of \(D\) is a set of volume zero.
If \(D\) is a bounded Jordan measurable set, we denote \[ \begin{aligned} \mu(D) = \int_D 1_D(x) d\mu, \end{aligned} \] which is called the \(m\)-dimensional volume of \(D\).
Theorem 5.3 Let \(D_1 \subset \mathbb{R}^m\) be a bounded closed Jordan measurable set, and \(a, b : D_1 \to \mathbb{R}\) be continuous functions. Then \[ \begin{aligned} D = \{(\mathbf{x}, y) \in \mathbb{R}^{m+1} \mid \mathbf{x} \in D_1, a(\mathbf{x}) \le y \le b(\mathbf{x})\} \end{aligned} \] is a bounded closed Jordan measurable set in \(\mathbb{R}^{m+1}\).
Theorem 5.4 Let \(D \subset \mathbb{R}^m\) be a bounded Jordan measurable set. Then:
A function \(f : D \to \mathbb{R}\) is Riemann integrable if and only if \(f\) is bounded and the set of discontinuities of \(f\) is a set of volume zero.
If \(D\) is also a closed set, then all continuous functions on \(D\) are Riemann integrable functions.
Theorem 5.5 If the function \(f\) is Riemann integrable on the bounded Jordan measurable sets \(D_1\) and \(D_2\), and \(D_1 \cap D_2 \subseteq \partial D_1 \cup \partial D_2\), then \(f\) is also Riemann integrable on \(D_1 \cup D_2\), and \[ \begin{aligned} \int_{D_1 \cup D_2} f(x) d\mu = \int_{D_1} f(x) d\mu + \int_{D_2} f(x) d\mu. \end{aligned} \]
5.3 Calculation of Multiple Integrals
Theorem 5.6 If \(f : [a, b] \times [\alpha, \beta] \to \mathbb{R}\) is a continuous function, then \[ \begin{aligned} \int_{[a, b] \times [\alpha, \beta]} f(x, y) dx dy = \int_{\alpha}^{\beta} \left( \int_{a}^{b} f(x, y) dx \right) dy = \int_{a}^{b} \left( \int_{\alpha}^{\beta} f(x, y) dy \right) dx. \end{aligned} \]
Theorem 5.7 Let \(R_1, R_2\) be rectangles in \(\mathbb{R}^m\) and \(\mathbb{R}^n\), respectively. Let \(f : R_1 \times R_2 \to \mathbb{R}\) be a Riemann integrable function. Then for any \(y \in R_2\), the Darboux upper integral and Darboux lower integral of \(f(x, y)\) with respect to \(x\): \[ \begin{aligned} \overline{\int_{R_1}} f(x, y) dx, \quad \underline{\int_{R_1}} f(x, y) dx \end{aligned} \] are both Riemann integrable functions (with respect to \(y\)), and \[ \begin{aligned} \int_{R_1 \times R_2} f(x, y) dx dy = \int_{R_2} \left( \overline{\int_{R_1}} f(x, y) dx \right) dy = \int_{R_2} \left( \underline{\int_{R_1}} f(x, y) dx \right) dy. \end{aligned} \]
Theorem 5.8 If \(D \subset \mathbb{R}^{m+n}\) is a bounded closed Jordan measurable set, then: \[ \begin{aligned} D_2 = \{y \in \mathbb{R}^n \mid \exists x \in \mathbb{R}^m \text{ s.t. } (x, y) \in D\} \end{aligned} \]
is a bounded closed Jordan measurable set in \(\mathbb{R}^n\), and for almost all \(y \in D_2\),
\[ \begin{aligned} D_y = \{x \in \mathbb{R}^m \mid (x, y) \in D\} \end{aligned} \]
is a bounded closed Jordan measurable set in \(\mathbb{R}^m\). For any continuous function \(f : D \to \mathbb{R}\),
\[ \begin{aligned} \int_D f(x, y) dx dy = \int_{D_2} \left( \int_{D_y} f(x, y) dx \right) dy. \end{aligned} \]
5.4 Calculation of Multiple Integrals: Change of Variables Formula
Theorem 5.9 Let \(\tilde{\Omega} \subset \mathbb{R}^m\) be a bounded closed Jordan measurable set, and \(\varphi : \tilde{\Omega} \to \Omega = \varphi(\tilde{\Omega}) \subseteq \mathbb{R}^m\) be a \(\mathscr{C}^1\) diffeomorphism. Then for any \(f \in \mathscr{R}(\Omega)\), \[ \begin{aligned} \int_{\Omega} f(u) d\mu_{\Omega} = \int_{\tilde{\Omega}} f(\varphi(x)) |\det D\varphi(x)| d\mu_{\tilde{\Omega}}. \end{aligned} \] In traditional notation, the above equation is expressed as: \[ \begin{aligned} \int_{\Omega} f(u) du_1 \dots du_m = \int_{\tilde{\Omega}} f(\varphi(x)) \left| \det \frac{\partial(u_1, \dots, u_m)}{\partial(x_1, \dots, x_m)} \right| dx_1 \dots dx_m. \end{aligned} \]
Theorem 5.10 Let \(\Sigma\) be an \(m \times m\) symmetric positive-definite matrix, and let \(\mu \in \mathbb{R}^m\). The integral of the multivariate Gaussian density function over the entire space \(\mathbb{R}^m\) is equal to 1: \[ \begin{aligned} \int_{\mathbb{R}^m} \frac{1}{\sqrt{(2\pi)^m \det \Sigma}} e^{-\frac{1}{2}(\mathbf{x}-\mu)^T \Sigma^{-1}(\mathbf{x}-\mu)} dx_1 \dots dx_m = 1. \end{aligned} \]