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 $\int_{C} F \cdot T d s,$ which we also denote $\int_{C} F \cdot T d s$

closed curve: a curve for which there exists a parameterization $r (t),$ $a \leq t \leq b,$ such that $r (a) = r (b),$ and the curve is traversed exactly once

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 $∇ f = F$

curl: the curl of vector field $F = 〈 P, Q, R 〉,$ denoted $\nabla \times F,$ is the “determinant” of the matrix $| \begin{matrix} i & j & k \\ \frac{\partial}{\partial x} & \frac{\partial}{\partial y} & \frac{\partial}{\partial z} \\ P & Q & R \end{matrix} |$ and is given by the expression $(R_{y} - Q_{z}) i + (P_{z} - R_{x}) j + (Q_{x} - P_{y}) k;$ 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 $F = 〈 P, Q, R 〉,$ denoted $\nabla \cdot F,$ 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 $\int_{C} F \cdot \frac{n (t)}{‖ n (t) ‖} d s$

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 $\int_{C} ∇ f \cdot d r$ depends only on the value of $f$ at the endpoints of C: $\int_{C} ∇ f \cdot d r = f (r (b)) - f (r (a))$

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 $Q / ε_{0}$

gradient field: a vector field $F$ for which there exists a scalar function $f$ such that $∇ f = F;$ 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 $\int_{C_{1}} F \cdot d r = \int_{C_{2}} F \cdot d r$ 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 $r (u, v) = 〈 x (u, v), y (u, v), z (u, v) 〉$

parameterized surface (parametric surface): a surface given by a description of the form $r (u, v) = 〈 x (u, v), y (u, v), z (u, v) 〉,$ 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

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 $r (u, v) = 〈 x (u, v), y (u, v), z (u, v) 〉$ such that $r_{u} \times r_{v}$ is not zero for any point $(u, v)$ in the parameter domain

rotational field: a vector field in which the vector at point $(x, y)$ 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 $\int_{C} f d s,$ 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

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 $F = 〈 P, Q 〉$ is a source-free vector field, then stream function g is a function such that $P = g_{y}$ and $Q = − g_{x}$

surface area: the area of surface S given by the surface integral $\int \int_{S} d S$

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 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 $ℝ^{2},$ an assignment of a vector $F (x, y)$ to each point $(x, y)$ of a subset $D$ of $ℝ^{2};$ in $ℝ^{3},$ an assignment of a vector $F (x, y, z)$ to each point $(x, y, z)$ of a subset $D$ of $ℝ^{3}$

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, $\int_{C} F \cdot T d s;$ such an integral is defined in terms of a Riemann sum, similar to a single-variable integral