double integral: of the function $f (x, y)$ over the region $R$ in the $x y$ -plane is defined as the limit of a double Riemann sum, $\iint_{R} f (x, y) d A = lim_{m, n \to \infty} \sum_{i = 1}^{m} \sum_{j = 1}^{n} f (x_{i j}^{*}, y_{i j}^{*}) Δ A .$

double Riemann sum: of the function $f (x, y)$ over a rectangular region $R$ is $\sum_{i = 1}^{m} \sum_{j = 1}^{n} f (x_{i j}^{*}, y_{i j}^{*}) Δ A$ where $R$ is divided into smaller subrectangles $R_{i j}$ and $(x_{i j}^{*}, y_{i j}^{*})$ is an arbitrary point in $R_{i j}$

Fubini’s theorem: if $f (x, y)$ is a function of two variables that is continuous over a rectangular region $R = {(x, y) \in ℝ^{2} | a \leq x \leq b, c \leq y \leq d},$ then the double integral of $f$ over the region equals an iterated integral, $\iint_{R} f (x, y) d y d x = \int_{a}^{b} \int_{c}^{d} f (x, y) d x d y = \int_{c}^{d} \int_{a}^{b} f (x, y) d x d y$

improper double integral: a double integral over an unbounded region or of an unbounded function

iterated integral

for a function

f (x, y)

over the region

R

is

$\int_{a}^{b} \int_{c}^{d} f (x, y) d x d y = \int_{a}^{b} [\int_{c}^{d} f (x, y) d y] d x,$
$\int_{c}^{d} \int_{b}^{a} f (x, y) d x d y = \int_{c}^{d} [\int_{a}^{b} f (x, y) d x] d y,$

where

a, b, c,

and

d

are any real numbers and

R = [a, b] \times [c, d]

Jacobian: the Jacobian $J (u, v)$ in two variables is a $2 \times 2$ determinant:

$J (u, v) = | \begin{array}{l} \frac{\partial x}{\partial u} & \frac{\partial y}{\partial u} \\ \frac{\partial x}{\partial v} & \frac{\partial y}{\partial v} \end{array} |;$

the Jacobian $J (u, v, w)$ in three variables is a $3 \times 3$ determinant:

$J (u, v, w) = | \begin{matrix} \frac{\partial x}{\partial u} & \frac{\partial y}{\partial u} & \frac{\partial z}{\partial u} \\ \frac{\partial x}{\partial v} & \frac{\partial y}{\partial v} & \frac{\partial z}{\partial v} \\ \frac{\partial x}{\partial w} & \frac{\partial y}{\partial w} & \frac{\partial z}{\partial w} \end{matrix} |$

one-to-one transformation: a transformation $T : G \to R$ defined as $T (u, v) = (x, y)$ is said to be one-to-one if no two points map to the same image point

planar transformation: a function $T$ that transforms a region $G$ in one plane into a region $R$ in another plane by a change of variables

polar rectangle: the region enclosed between the circles $r = a$ and $r = b$ and the angles $θ = α$ and $θ = β;$ it is described as $R = {(r, θ) | a \leq r \leq b, α \leq θ \leq β}$

radius of gyration: the distance from an object’s center of mass to its axis of rotation

transformation: a function that transforms a region $G$ in one plane into a region $R$ in another plane by a change of variables

triple integral: the triple integral of a continuous function $f (x, y, z)$ over a rectangular solid box $B$ is the limit of a Riemann sum for a function of three variables, if this limit exists

triple integral in cylindrical coordinates: the limit of a triple Riemann sum, provided the following limit exists:

$lim_{l, m, n \to \infty} \sum_{i = 1}^{l} \sum_{j = 1}^{m} \sum_{k = 1}^{n} f (r_{i j k}^{*}, θ_{i j k}^{*}, z_{i j k}^{*}) r_{i j k}^{*} Δ r Δ θ Δ z$

triple integral in spherical coordinates: the limit of a triple Riemann sum, provided the following limit exists:

$lim_{l, m, n \to \infty} \sum_{i = 1}^{l} \sum_{j = 1}^{m} \sum_{k = 1}^{n} f (ρ_{i j k}^{*}, θ_{i j k}^{*}, φ_{i j k}^{*}) {(ρ_{i j k}^{*})}^{2} sin φ Δ ρ Δ θ Δ φ$

Type I: a region $D$ in the $x y$ -plane is Type I if it lies between two vertical lines and the graphs of two continuous functions $g_{1} (x)$ and $g_{2} (x)$

Type II: a region $D$ in the $x y$ -plane is Type II if it lies between two horizontal lines and the graphs of two continuous functions $h_{1} (y) and h_{2} (y)$