Key Terms

double integral

of the function $f(x,y)$ over the region $R$ in the $xy$-plane is defined as the limit of a double Riemann sum, ${\displaystyle \underset{R}{\iint }f(x,y)dA}=\underset{m,n\to \infty }{\text{lim}}{\displaystyle \sum _{i=1}^m{\displaystyle \sum _{j=1}^{n}f(x_{ij}^{*},y_{ij}^{*})\text{Δ}A.}}$

double Riemann sum

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

Fubini’s theorem

if $f(x,y)$ is a function of two variables that is continuous over a rectangular region $R=\{(x,y)\in {ℝ}^2|a\le x\le b,c\le y\le d\},$ then the double integral of $f$ over the region equals an iterated integral, ${\displaystyle \underset{R}{\iint }f(x,y)dydx}={\displaystyle {\int }_a^b{\displaystyle {\int }_c^{d}f(x,y)dxdy}}={\displaystyle {\int }_c^d{\displaystyle {\int }_a^{b}f(x,y)dxdy}}$

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

1. ${\displaystyle {\int }_a^b{\displaystyle {\int }_c^{d}f(x,y)dxdy}}={\displaystyle {\int }_a^b\left[{\displaystyle {\int }_c^{d}f(x,y)dy}\right]}dx,$
2. ${\displaystyle {\int }_c^d{\displaystyle {\int }_b^{a}f(x,y)dxdy}}={\displaystyle {\int }_c^d\left[{\displaystyle {\int }_a^{b}f(x,y)dx}\right]}dy,$

where $a,b,c,$ and $d$ are any real numbers and $R=[a,b]\times [c,d]$

Jacobian

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

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

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

$$J(u,v,w)=\left|\begin{array}{ccccc}\frac{\partial x}{\partial u}\hfill & & \frac{\partial y}{\partial u}\hfill & & \frac{\partial z}{\partial u}\hfill \\ \frac{\partial x}{\partial v}\hfill & & \frac{\partial y}{\partial v}\hfill & & \frac{\partial z}{\partial v}\hfill \\ \frac{\partial x}{\partial w}\hfill & & \frac{\partial y}{\partial w}\hfill & & \frac{\partial z}{\partial w}\hfill \end{array}\right|$$

one-to-one transformation

a transformation $T:G\to R$ defined as $T\left(u,v\right)=\left(x,y\right)$ 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 $\theta =\alpha$ and $\theta =\beta ;$ it is described as $R=\left\{\left(r,\theta \right)|a\le r\le b,\alpha \le \theta \le \beta \right\}$

radius of gyration

the distance between the rotational axis of the object and the point where the entire mass of the object can be concentrated and have the same moment of inertia

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 $f\left(x,y,z\right)$ 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:

$$\underset{l,m,n\to \infty }{\text{lim}}{\displaystyle \sum _{i=1}^l{\displaystyle \sum _{j=1}^m{\displaystyle \sum _{k=1}^{n}f(r_{ijk}^{*},{\theta }_{ijk}^{*},z_{ijk}^{*})r_{ijk}^{*}\text{Δ}r\text{Δ}\theta \text{Δ}z}}}$$

triple integral in spherical coordinates

the limit of a triple Riemann sum, provided the following limit exists:

$$\underset{l,m,n\to \infty }{\text{lim}}{\displaystyle \sum _{i=1}^l{\displaystyle \sum _{j=1}^m{\displaystyle \sum _{k=1}^{n}f({\rho }_{ijk}^{*},{\theta }_{ijk}^{*},{\phi }_{ijk}^{*}){({\rho }_{ijk}^{*})}^2\text{sin}\phi \text{Δ}\rho \text{Δ}\theta \text{Δ}\phi }}}$$

Type I

a region $D$ in the $xy$-plane is Type I if it lies between two vertical lines and the graphs of two continuous functions $g_1\left(x\right)$ and $g_2\left(x\right)$

Type II

a region $D$ in the $xy$-plane is Type II if it lies between two horizontal lines and the graphs of two continuous functions $h_1\left(y\right)\text{and}{h}_2\left(y\right)$