Key Terms

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 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 $f\left(x,y,z\right)$ 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 $h_1\left(y\right)\text{and}{h}_2\left(y\right)$