Chapter 6: Vector Calculus

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·𝐓⁢𝑑⁢𝑠,

closed curve

a curve for which there exists a parameterization 𝐫(𝑡), 𝑎≤𝑡≤𝑏, such that 𝐫(𝑎)=𝐫⁡(𝑏), and the curve is traversed exactly once; 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/ gradient field

a vector field for which there exists a scalar function 𝑓 such that ∇𝑓=𝐅

curl

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

of a vector field 𝐅=〈𝑃,𝑄,𝑅〉, denoted ∇⋅𝐅, is 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;

flux integral

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

Fundamental Theorem for Line Integrals

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/𝜀0

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

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 C

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

potential function

a scalar function 𝑓 such that ∇𝑓=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 𝐫(𝑢,𝑣)=〈𝑥⁡(𝑢,𝑣),𝑦⁡(𝑢,𝑣),𝑧⁡(𝑢,𝑣)〉 such that 𝐫𝑢×𝐫𝑣 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=√𝑥²+𝑦²; all vectors flow either clockwise or counterclockwise, and the magnitude of a vector depends only on its distance from the origin

scalar line integral

of a function 𝑓 along a curve C with respect to arc length is the integral ∫𝐶𝑓⁡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

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〉 is a source-free vector field, then g is a function such that 𝑃=𝑔𝑦 and 𝑄=−𝑔𝑥

surface area

the area of surface S given by the surface integral

surface independent

flux integrals of curl vector fields 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, an assignment of a vector 𝐅(𝑥,𝑦) to each point (𝑥,𝑦) of a subset 𝐷 of ℝ²; in ℝ³, an assignment of a vector 𝐅(𝑥,𝑦,𝑧) to each point (𝑥,𝑦,𝑧) of a subset 𝐷 of ℝ³

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·𝐓⁢𝑑⁢𝑠; such an integral is defined in terms of a Riemann sum, similar to a single-variable integral

Circulation of a conservative field over curve C that encloses a simply connected region

Green’s theorem, circulation form

Green’s theorem, flux form

Green’s theorem, extended version

Divergence of curl is zero

Curl of a gradient is the zero vector

Scalar Surface Integral

Calculating Flux