### Key Terms

- circulation
- the tendency of a fluid to move in the direction of curve
*C*. If*C*is a closed curve, then the circulation of**F**along*C*is line integral ${\int}_{C}\text{F}\xb7\text{T}ds,$ which we also denote ${\oint}_{C}\text{F}\xb7\text{T}ds$

- closed curve
- a curve for which there exists a parameterization $\text{r}\left(t\right),$ $a\le t\le b,$ such that $\text{r}\left(a\right)=\text{r}\left(b\right),$ and the curve is traversed exactly once

- closed curve
- a curve that begins and ends at the same point

- connected region
- a region in which any two points can be connected by a path with a trace contained entirely inside the region

- conservative field
- a vector field for which there exists a scalar function $f$ such that $\text{\u2207}f=\text{F}$

- curl
- the curl of vector field $\text{F}=\langle P,Q,R\rangle ,$ denoted $\nabla \phantom{\rule{0.2em}{0ex}}\times \phantom{\rule{0.2em}{0ex}}\text{F},$ is the “determinant” of the matrix $\left|\begin{array}{ccc}\hfill \text{i}\hfill & \hfill \text{j}\hfill & \hfill \text{k}\hfill \\ \hfill \frac{\partial}{\partial x}\hfill & \hfill \frac{\partial}{\partial y}\hfill & \hfill \frac{\partial}{\partial z}\hfill \\ \hfill P\hfill & \hfill Q\hfill & \hfill R\hfill \end{array}\right|$ and is given by the expression $\left({R}_{y}-{Q}_{z}\right)\text{i}+\left({P}_{z}-{R}_{x}\right)\text{j}+\left({Q}_{x}-{P}_{y}\right)\text{k};$ it measures the tendency of particles at a point to rotate about the axis that points in the direction of the curl at the point

- divergence
- the divergence of a vector field $\text{F}=\langle P,Q,R\rangle ,$ denoted $\nabla \phantom{\rule{0.2em}{0ex}}\cdot \phantom{\rule{0.2em}{0ex}}\text{F},$ is ${P}_{x}+{Q}_{y}+{R}_{z};$ it measures the “outflowing-ness” of a vector field

- divergence theorem
- a theorem used to transform a difficult flux integral into an easier triple integral and vice versa

- flux
- the rate of a fluid flowing across a curve in a vector field; the flux of vector field
**F**across plane curve*C*is line integral ${\int}_{C}\text{F}\xb7\frac{\text{n}(t)}{\Vert \text{n}(t)\Vert}}\phantom{\rule{0.2em}{0ex}}ds$

- flux integral
- another name for a surface integral of a vector field; the preferred term in physics and engineering

- Fundamental Theorem for Line Integrals
- the value of line integral $\int}_{C}\text{\u2207}f\xb7d\text{r$ depends only on the value of $f$ at the endpoints of
*C*: ${\int}_{C}\text{\u2207}f\xb7d\text{r}}=f\left(\text{r}(b\right))-f\left(\text{r}(a)\right)$

- Gauss’ law
- if
*S*is a piecewise, smooth closed surface in a vacuum and*Q*is the total stationary charge inside of*S*, then the flux of electrostatic field**E**across*S*is $Q\text{/}{\epsilon}_{0}$

- gradient field
- a vector field $\text{F}$ for which there exists a scalar function $f$ such that $\text{\u2207}f=\text{F};$ in other words, a vector field that is the gradient of a function; such vector fields are also called
*conservative*

- Green’s theorem
- relates the integral over a connected region to an integral over the boundary of the region

- grid curves
- curves on a surface that are parallel to grid lines in a coordinate plane

- heat flow
- a vector field proportional to the negative temperature gradient in an object

- independence of path
- a vector field
**F**has path independence if $\int}_{{C}_{1}}\text{F}\xb7d\text{r}}={\displaystyle {\int}_{{C}_{2}}\text{F}\xb7d\text{r$ for any curves ${C}_{1}$ and ${C}_{2}$ in the domain of**F**with the same initial points and terminal points

- inverse-square law
- the electrostatic force at a given point is inversely proportional to the square of the distance from the source of the charge

- line integral
- the integral of a function along a curve in a plane or in space

- mass flux
- the rate of mass flow of a fluid per unit area, measured in mass per unit time per unit area

- orientation of a curve
- the orientation of a curve
*C*is a specified direction of*C*

- orientation of a surface
- if a surface has an “inner” side and an “outer” side, then an orientation is a choice of the inner or the outer side; the surface could also have “upward” and “downward” orientations

- parameter domain (parameter space)
- the region of the
*uv*plane over which the parameters*u*and*v*vary for parameterization $\text{r}\left(u,v\right)=\langle x\left(u,v\right),y\left(u,v\right),z\left(u,v\right)\rangle $

- parameterized surface (parametric surface)
- a surface given by a description of the form $\text{r}\left(u,v\right)=\langle x\left(u,v\right),y\left(u,v\right),z\left(u,v\right)\rangle ,$ where the parameters
*u*and*v*vary over a parameter domain in the*uv*-plane

- piecewise smooth curve
- an oriented curve that is not smooth, but can be written as the union of finitely many smooth curves

- potential function
- a scalar function $f$ such that $\text{\u2207}f=\text{F}$

- radial field
- a vector field in which all vectors either point directly toward or directly away from the origin; the magnitude of any vector depends only on its distance from the origin

- regular parameterization
- parameterization $\text{r}(u,v)=\langle x(u,v),y(u,v),z(u,v)\rangle $ such that ${\text{r}}_{u}\phantom{\rule{0.2em}{0ex}}\times \phantom{\rule{0.2em}{0ex}}{\text{r}}_{v}$ is not zero for point $(u,v)$ in the parameter domain

- rotational field
- a vector field in which the vector at point $\left(x,y\right)$ is tangent to a circle with radius $r=\sqrt{{x}^{2}+{y}^{2}};$ in a rotational field, all vectors flow either clockwise or counterclockwise, and the magnitude of a vector depends only on its distance from the origin

- scalar line integral
- the scalar line integral of a function $f$ along a curve
*C*with respect to arc length is the integral ${\int}_{C}fds},$ it is the integral of a scalar function $f$ along a curve in a plane or in space; such an integral is defined in terms of a Riemann sum, as is a single-variable integral

- simple curve
- a curve that does not cross itself

- simply connected region
- a region that is connected and has the property that any closed curve that lies entirely inside the region encompasses points that are entirely inside the region

- Stokes’ theorem
- relates the flux integral over a surface
*S*to a line integral around the boundary*C*of the surface*S*

- stream function
- if $\text{F}=\langle P,Q\rangle $ is a source-free vector field, then stream function
*g*is a function such that $P={g}_{y}$ and $Q=\text{\u2212}{g}_{x}$

- surface area
- the area of surface
*S*given by the surface integral $\int {\displaystyle {\int}_{S}dS}$

- surface independent
- flux integrals of curl vector fields are surface independent if their evaluation does not depend on the surface but only on the boundary of the surface

- surface integral
- an integral of a function over a surface

- surface integral of a scalar-valued function
- a surface integral in which the integrand is a scalar function

- surface integral of a vector field
- a surface integral in which the integrand is a vector field

- unit vector field
- a vector field in which the magnitude of every vector is 1

- vector field
- measured in ${\mathbb{R}}^{2},$ an assignment of a vector $\text{F}\left(x,y\right)$ to each point $\left(x,y\right)$ of a subset $D$ of ${\mathbb{R}}^{2};$ in ${\mathbb{R}}^{3},$ an assignment of a vector $\text{F}\left(x,y,z\right)$ to each point $\left(x,y,z\right)$ of a subset $D$ of ${\mathbb{R}}^{3}$

- vector line integral
- the vector line integral of vector field
**F**along curve*C*is the integral of the dot product of**F**with unit tangent vector**T**of*C*with respect to arc length, ${\int}_{C}\text{F}\xb7\text{T}ds};$ such an integral is defined in terms of a Riemann sum, similar to a single-variable integral