Calculus Volume 3

# Key Terms

### 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 $∫CF·Tds,∫CF·Tds,$ which we also denote $∮CF·Tds∮CF·Tds$
closed curve
a curve for which there exists a parameterization $r(t),r(t),$ $a≤t≤b,a≤t≤b,$ such that $r(a)=r(b),r(a)=r(b),$ 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 $ff$ such that $∇f=F∇f=F$
curl
the curl of vector field $F=〈P,Q,R〉,F=〈P,Q,R〉,$ denoted $∇×F,∇×F,$ is the “determinant” of the matrix $|ijk∂∂x∂∂y∂∂zPQR||ijk∂∂x∂∂y∂∂zPQR|$ and is given by the expression $(Ry−Qz)i+(Pz−Rx)j+(Qx−Py)k;(Ry−Qz)i+(Pz−Rx)j+(Qx−Py)k;$ 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 $F=〈P,Q,R〉,F=〈P,Q,R〉,$ denoted $∇×F,∇×F,$ is $Px+Qy+Rz;Px+Qy+Rz;$ 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 $∫CF·n(t)‖n(t)‖ds∫CF·n(t)‖n(t)‖ds$
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 $∫C∇f·dr∫C∇f·dr$ depends only on the value of $ff$ at the endpoints of C: $∫C∇f·dr=f(r(b))−f(r(a))∫C∇f·dr=f(r(b))−f(r(a))$
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 $Q/ε0Q/ε0$
a vector field $FF$ for which there exists a scalar function $ff$ such that $∇f=F;∇f=F;$ 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 $∫C1F·dr=∫C2F·dr∫C1F·dr=∫C2F·dr$ for any curves $C1C1$ and $C2C2$ 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 $r(u,v)=〈x(u,v),y(u,v),z(u,v)〉r(u,v)=〈x(u,v),y(u,v),z(u,v)〉$
parameterized surface (parametric surface)
a surface given by a description of the form $r(u,v)=〈x(u,v),y(u,v),z(u,v)〉,r(u,v)=〈x(u,v),y(u,v),z(u,v)〉,$ 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
potential function
a scalar function $ff$ such that $∇f=F∇f=F$
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 $r(u,v)=〈x(u,v),y(u,v),z(u,v)〉r(u,v)=〈x(u,v),y(u,v),z(u,v)〉$ such that $ru×rvru×rv$ is not zero for point $(u,v)(u,v)$ in the parameter domain
rotational field
a vector field in which the vector at point $(x,y)(x,y)$ is tangent to a circle with radius $r=x2+y2;r=x2+y2;$ 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 $ff$ along a curve C with respect to arc length is the integral $∫Cfds,∫Cfds,$ it is the integral of a scalar function $ff$ 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 $F=〈P,Q〉F=〈P,Q〉$ is a source-free vector field, then stream function g is a function such that $P=gyP=gy$ and $Q=−gxQ=−gx$
surface area
the area of surface S given by the surface integral $∫∫SdS∫∫SdS$
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 $ℝ2,ℝ2,$ an assignment of a vector $F(x,y)F(x,y)$ to each point $(x,y)(x,y)$ of a subset $DD$ of $ℝ2;ℝ2;$ in $ℝ3,ℝ3,$ an assignment of a vector $F(x,y,z)F(x,y,z)$ to each point $(x,y,z)(x,y,z)$ of a subset $DD$ of $ℝ3ℝ3$
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, $∫CF·Tds;∫CF·Tds;$ such an integral is defined in terms of a Riemann sum, similar to a single-variable integral
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