Multiple Integrals

Flashcards covering double and triple integrals, properties, Fubini's theorem, change of variables, cylindrical and spherical coordinates, and applications.

Double Integral (Riemann Sum Definition)

The double integral of a function $f$ on a domain $\mathcal{D}$ is the limit $\lim_{n \to +\infty} \sum_{i=1}^n f(x_i, y_i) (\delta x \delta y)_i$, denoted by $\iint_\mathcal{D} f(x, y)\,dx\,dy$.

Integrability Theorem for Double Integrals

If $f$ is a continuous function on the domain $\mathcal{D}$, and $\mathcal{D}$ is bounded, then $f$ is integrable on $\mathcal{D}$.

Positivity Property of Double Integrals

If $f(x, y) \ge 0$ for all $(x, y) \in \mathcal{D}$, then $\iint_\mathcal{D} f(x, y)\,dx\,dy \ge 0$.

Monotonicity Property of Double Integrals

If $f(x, y) \le g(x, y)$ for all $(x, y) \in \mathcal{D}$, then $\iint_\mathcal{D} f(x, y)\,dx\,dy \le \iint_\mathcal{D} g(x, y)\,dx\,dy$.

Linearity Property of Double Integrals

For any constants $\alpha, \beta \in \mathbb{R}$, $\iint_\mathcal{D} [\alpha f(x, y) + \beta g(x, y)]\,dx\,dy = \alpha \iint_\mathcal{D} f(x, y)\,dx\,dy + \beta \iint_\mathcal{D} g(x, y)\,dx\,dy$.

Domain Additivity Property of Double Integrals

If $\mathcal{D} = \mathcal{D}_1 \cup \mathcal{D}_2$ where $\mathcal{D}_1 \cap \mathcal{D}_2$ has zero area, then $\iint_\mathcal{D} f(x, y)\,dx\,dy = \iint_{\mathcal{D}_1} f(x, y)\,dx\,dy + \iint_{\mathcal{D}_2} f(x, y)\,dx\,dy$.

Fubini's Theorem (Type I Domain in 2D)

For a domain $\mathcal{D} = \{(x, y) \in \mathbb{R}^2 : a \le x \le b, \alpha_1(x) \le y \le \alpha_2(x)\}$ where $\alpha_1, \alpha_2$ are continuous on $[a, b]$, $\iint_\mathcal{D} f(x, y)\,dx\,dy = \int_a^b \left( \int_{\alpha_1(x)}^{\alpha_2(x)} f(x, y)\,dy \right) dx$.

Fubini's Theorem (Type II Domain in 2D)

For a domain $\mathcal{D} = \{(x, y) \in \mathbb{R}^2 : c \le y \le d, \beta_1(y) \le x \le \beta_2(y)\}$ where $\beta_1, \beta_2$ are continuous on $[c, d]$, $\iint_\mathcal{D} f(x, y)\,dx\,dy = \int_c^d \left( \int_{\beta_1(y)}^{\beta_2(y)} f(x, y)\,dx \right) dy$.

Double Integral of a Separable Variable Function

If $f(x, y) = g(x) h(y)$ and $\mathcal{D} = [a, b] \times [c, d]$, then $\iint_\mathcal{D} f(x, y)\,dx\,dy = \left[ \int_a^b g(x)\,dx \right] \cdot \left[ \int_c^d h(y)\,dy \right]$.

Change of Variables in Double Integrals

For $x = \phi(u, v)$ and $y = \Psi(u, v)$, $\iint_\mathcal{D} f(x, y)\,dx\,dy = \iint_\Delta f(\phi(u, v), \Psi(u, v)) |J|\,du\,dv$, where $J = \frac{\partial (x, y)}{\partial (u, v)}$.

2D Jacobian Determinant

The determinant of partial derivatives for the coordinate change $x = \phi(u, v)$ and $y = \Psi(u, v)$, given by $J = \frac{\partial (x, y)}{\partial (u, v)} = x_u y_v - x_v y_u$.

Polar Coordinates Transformation

The transformation $x = r \cos(\theta)$ and $y = r \sin(\theta)$ with Jacobian $J = r$, converting double integrals to $\iint_\mathcal{D} f(x, y)\,dx\,dy = \iint_\Delta f(r \cos(\theta), r \sin(\theta)) r\,dr\,d\theta$.

Area of a Planar Domain

The total area of a two-dimensional domain $\mathcal{D}$, given by $\text{area}(\mathcal{D}) = \iint_\mathcal{D} dx\,dy$.

Mass of a Planar Lamina

The total mass $M$ of a lamina in domain $\mathcal{D}$ with mass density $\rho(x, y)$, calculated as $M = \iint_\mathcal{D} \rho(x, y)\,dx\,dy$.

Center of Mass of a Planar Lamina

The center of mass coordinates $(x_G, y_G)$ given by $x_G = \frac{1}{M} \iint_\mathcal{D} x \rho(x, y)\,dx\,dy$ and $y_G = \frac{1}{M} \iint_\mathcal{D} y \rho(x, y)\,dx\,dy$.

Triple Integral (Riemann Sum Definition)

The triple integral of a function $f$ on a domain $\Omega \subset \mathbb{R}^3$ is $\lim_{n \to +\infty} \sum_{i=1}^n f(x_i, y_i, z_i) \bar{\omega}_i$, denoted by $\iiint_\Omega f(x, y, z)\,dx\,dy\,dz$.

Integrability Theorem for Triple Integrals

If $f$ is a continuous function on the domain $\Omega$, and $\Omega$ is bounded in $\mathbb{R}^3$, then $f$ is integrable on $\Omega$.

Fubini's Theorem (Projection Method for Triple Integrals)

For a domain $\Omega = \{(x, y, z) \in \mathbb{R}^3 : (x, y) \in \mathcal{D}, \Psi_1(x, y) \le z \le \Psi_2(x, y)\}$ where $\mathcal{D}$ is the projection of $\Omega$ onto the $xy$-plane, $\iiint_\Omega f(x, y, z)\,dx\,dy\,dz = \iint_\mathcal{D} \left( \int_{\Psi_1(x, y)}^{\Psi_2(x, y)} f(x, y, z)\,dz \right) dx\,dy$.

Change of Variables in Triple Integrals

For transformation $(x, y, z) = \varphi(u, v, w)$, $\iiint_\Omega f(x, y, z)\,dx\,dy\,dz = \iiint_\Delta f(\varphi(u, v, w)) |J|\,du\,dv\,dw$, where $J = \frac{\partial (x, y, z)}{\partial (u, v, w)}$.

3D Jacobian Determinant

The determinant of partial derivatives for the coordinate change $(x, y, z) = \varphi(u, v, w)$, denoted by $J = \frac{\partial (x, y, z)}{\partial (u, v, w)}$.

Cylindrical Coordinates Transformation

The coordinate system defined by $x = r \cos(\theta)$, $y = r \sin(\theta)$, $z = z$, with Jacobian $J = r$, giving $\iiint_\Omega f(x, y, z)\,dx\,dy\,dz = \iiint_\Omega f(r \cos(\theta), r \sin(\theta), z) r\,dr\,d\theta\,dz$.

Cone-Bounded Solid in Cylindrical Coordinates

A solid domain $\Omega = \{(r, \theta, z) : 0 \le \theta \le 2\pi, 0 \le r \le 2, r \le z \le 2\}$ formed between the cone $z = \sqrt{x^2 + y^2}$ and plane $z = 2$.

Spherical Coordinates Transformation

The coordinate system defined by $x = r \cos(\theta) \sin(\varphi)$, $y = r \sin(\theta) \sin(\varphi)$, $z = r \cos(\varphi)$, with Jacobian determinant $J = -r^2 \sin(\varphi)$ and absolute value $|J| = r^2 \sin(\varphi)$, where $r \ge 0$, $0 \le \theta \le 2\pi$, $0 \le \varphi \le \pi$.

Volume of a 3D Domain

The total volume of a 3D domain $\Omega$, calculated as $\text{volume}(\Omega) = \iiint_\Omega dx\,dy\,dz$.