Essential Concepts

Introduction to Vector Fields

• A vector field assigns a vector [latex]{\bf{F}}(x,y)[/latex] to each point [latex](x,y)[/latex] in a subset [latex]D[/latex] of [latex]\mathbb{R}^{2}[/latex] or [latex]\mathbb{R}^{3}[/latex]. [latex]{\bf{F}}(x,y,z)[/latex] to each point [latex](x,y,z)[/latex] in a subset [latex]D[/latex] of [latex]\mathbb{R}^{3}[/latex].
• Vector fields can describe the distribution of vector quantities such as forces or velocities over a region of the plane or of space. They are in common use in such areas as physics, engineering, meteorology, oceanography.
• We can sketch a vector field by examining its defining equation to determine relative magnitudes in various locations and then drawing enough vectors to determine a pattern.
• A vector field [latex]{\bf{F}}[/latex] is called conservative if there exists a scalar function [latex]f[/latex] such that [latex]\nabla{f}={\bf{F}}[/latex].

Line Integrals

• Line integrals generalize the notion of a single-variable integral to higher dimensions. The domain of integration in a single-variable integral is a line segment along the x-axis, but the domain of integration in a line integral is a curve in a plane or in space.
• If [latex]C[/latex] is a curve, then the length of [latex]C[/latex] is [latex]\displaystyle\int_{C} ds[/latex].
• There are two kinds of line integral: scalar line integrals and vector line integrals. Scalar line integrals can be used to calculate the mass of a wire; vector line integrals can be used to calculate the work done on a particle traveling through a field.
• Scalar line integrals can be calculated using[latex]\displaystyle\int_{C} f(x,y,z)ds=\displaystyle\int_{a}^{b} f({\bf{r}}(t))\sqrt{(x^{\prime}(t))^{2}+(y^{\prime}(t))^{2}+(z^{\prime}(t))^{2}}dt[/latex]; vector line integrals can be calculated using [latex]\displaystyle\int_{C} {\bf{F}}\cdot{ds}=\displaystyle\int_{C}{\bf{F}}\cdot{\bf{T}}ds=\displaystyle\int_{a}^{b}{\bf{F}}({\bf{r}}(t))\cdot{\bf{r}}^{\prime}(t)dt[/latex].
• Two key concepts expressed in terms of line integrals are flux and circulation. Flux measures the rate that a field crosses a given line; circulation measures the tendency of a field to move in the same direction as a given closed curve.

Conservative Vector Fields

• The theorems in this section require curves that are closed, simple, or both, and regions that are connected or simply connected.
• The line integral of a conservative vector field can be calculated using the Fundamental Theorem for Line Integrals. This theorem is a generalization of the Fundamental Theorem of Calculus in higher dimensions. Using this theorem usually makes the calculation of the line integral easier.
• Conservative fields are independent of path. The line integral of a conservative field depends only on the value of the potential function at the endpoints of the domain curve.
• Given vector field [latex]{\bf{F}}[/latex], we can test whether [latex]{\bf{F}}[/latex] is conservative by using the cross-partial property. If [latex]{\bf{F}}[/latex] has the cross-partial property and the domain is simply connected, then [latex]{\bf{F}}[/latex] is conservative (and thus has a potential function). If [latex]{\bf{F}}[/latex] is conservative, we can find a potential function by using the Problem-Solving Strategy.
• The circulation of a conservative vector field on a simply connected domain over a closed curve is zero

Green’s Theorem

• Green’s theorem relates the integral over a connected region to an integral over the boundary of the region. Green’s theorem is a version of the Fundamental Theorem of Calculus in one higher dimension.
• Green’s Theorem comes in two forms: a circulation form and a flux form. In the circulation form, the integrand is [latex]{\bf{F}}\cdot{\bf{T}}[/latex]. In the flux form, the integrand is [latex]{\bf{F}}\cdot{\bf{N}}[/latex].
• Green’s theorem can be used to transform a difficult line integral into an easier double integral, or to transform a difficult double integral into an easier line integral.
• A vector field is source free if it has a stream function. The flux of a source-free vector field across a closed curve is zero, just as the circulation of a conservative vector field across a closed curve is zero.

Key Equations

• Vector field in [latex]\mathbb{R}^{2}[/latex]
  [latex]{\bf{F}}(x,y)=\langle{P(x,y),Q(x,y)}\rangle[/latex]  or  [latex]{\bf{F}}(x,y)=P(x,y){\bf{i}},Q(x,y){\bf{j}}[/latex]
• Vector field in [latex]\mathbb{R}^{3}[/latex]
  [latex]{\bf{F}}(x,y,z)=\langle{P(x,y,z),Q(x,y,z),R(x,y,z)}\rangle[/latex]  or  [latex]{\bf{F}}(x,y,z)=P(x,y,z){\bf{i}},Q(x,y,z){\bf{j}},R(x,y,z){\bf{k}}[/latex]
• Calculating a scalar line integral
  [latex]\displaystyle\int_{C} f(x,y,z)ds=\displaystyle\int_{a}^{b} f({\bf{r}}(t))\sqrt{(x^{\prime}(t))^{2}+(y^{\prime}(t))^{2}+(z^{\prime}(t))^{2}}dt[/latex]
• Calculating a vector line integral
  [latex]\displaystyle\int_{C} {\bf{F}}\cdot{ds}=\displaystyle\int_{C}{\bf{F}}\cdot{\bf{T}}ds=\displaystyle\int_{a}^{b}{\bf{F}}({\bf{r}}(t))\cdot{\bf{r}}^{\prime}(t)dt[/latex]
  or
  [latex]{\displaystyle\int_{C}} Pdx+Qdy+Rdz= {\displaystyle\int_{a}^{b}}\left(P({\bf{r}}(t))\frac{dx}{dt}+Q({\bf{r}}(t))\frac{dy}{dt}+R({\bf{r}}(t))\frac{dz}{dt}\right)dt[/latex]
• Calculating flux
  [latex]\displaystyle\int_{C} {\bf{F}}\cdot{\frac{{\bf{n}}(t)}{\Arrowvert{\bf{n}}(t)\Arrowvert}}ds=\displaystyle\int_{a}^{b}{\bf{F}}({\bf{r}}(t))\cdot{\bf{n}}(t)dt[/latex]
• Fundamental Theorem for Line Integrals
  [latex]\displaystyle\int_{C}\nabla {f}\cdot {d{\bf{r}}}=f({\bf{r}}(b))-f({\bf{r}}(a))[/latex]
• Circulation of a conservative field over curve [latex]C[/latex] that encloses a simply connected region 
  [latex]\displaystyle\oint_{C}\nabla {f}\cdot {d{\bf{r}}}=0[/latex]
• Green’s theorem, circulation form
  [latex]\displaystyle\oint_{C} Pdx+Qdy=\displaystyle\iint_{D} Q_{x}-P_{y}dA[/latex], where [latex]C[/latex] is the boundary of [latex]D[/latex]
• Green’s theorem, flux form
  [latex]\displaystyle\oint_{C} {\bf{F}}\cdot {d{\bf{r}}} = \displaystyle\iint_{D} Q_{x}-P_{y}dA[/latex], where [latex]C[/latex] is the boundary of [latex]D[/latex]
• Green’s theorem, extended version
  [latex]\displaystyle\oint_{\partial{D}} {\bf{F}}\cdot {d{\bf{r}}} = \displaystyle\iint_{D} Q_{x}-P_{y}dA[/latex]

Glossary

circulation

the tendency of a fluid to move in the direction of curve [latex]C[/latex]. If [latex]C[/latex] is a closed curve, then the circulation of [latex]{\bf{F}}[/latex] along [latex]C[/latex] is line integral [latex]\displaystyle\int_{C} {\bf{F}}\cdot{\bf{T}}ds[/latex], which we also denote [latex]\displaystyle\oint_{C} {\bf{F}}\cdot{\bf{T}}ds[/latex]

closed curve

a curve that begins and ends at the same point

closed curve

a curve for which there exists a parameterization [latex]{\bf{r}}(t),a\le{t}\le{b}[/latex], such that [latex]{\bf{r}}(a)={\bf{r}}(b)[/latex], 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 [latex]f[/latex] such that [latex]\nabla{f}={\bf{F}}[/latex]

flux

the rate of a fluid flowing across a curve in a vector field; the flux of vector field [latex]{\bf{F}}[/latex] across plane curve [latex]C[/latex] is line integral [latex]\displaystyle\int_{C} {\bf{F}}\cdot{\frac{{\bf{n}}(t)}{\Arrowvert{\bf{n}}(t)\Arrowvert}}ds[/latex]

Fundamental Theorem for Line Integrals

the value of the line integral [latex]\displaystyle\int_{C} {\nabla}{f}\cdot{d{\bf{r}}}[/latex] depends only on the value of [latex]f[/latex] at the endpoints of [latex]C[/latex]: [latex]\displaystyle\int_{C} {\nabla}{f}\cdot{d{\bf{r}}}=f({\bf{r}}(b)))-f({\bf{r}}(a))[/latex]

gradient field

a vector field [latex]{\bf{F}}[/latex] for which there exists a scalar function [latex]f[/latex] such that [latex]\nabla{f}={\bf{F}}[/latex] 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

independent of path (path independent)

a vector field [latex]{\bf{F}}[/latex] has path independence if [latex]\displaystyle\int_{C_{1}} {\bf{F}}\cdot{d{\bf{r}}}=\displaystyle\int_{C_{2}} {\bf{F}}\cdot{d{\bf{r}}}[/latex] for any curves [latex]C_{1}[/latex] and [latex]C_{2}[/latex] in the domain of [latex]{\bf{F}}[/latex] with the same initial points and terminal points

line integral

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

orientation of a curve

the orientation of a curve [latex]C[/latex] is a specified direction of [latex]C[/latex]

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 [latex]f[/latex] such that [latex]\nabla{f}={\bf{F}}[/latex]

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

rotational field

a vector field in which the vector at point [latex](x,y)[/latex] is tangent to a circle with radius [latex]r=\sqrt{x^{2}+y^{2}}[/latex] 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 [latex]f[/latex] along a curve [latex]C[/latex] with respect to arc length is the integral [latex]\displaystyle\int_C \! f\, \mathrm{d}s[/latex], it is the integral of a scalar function [latex]f[/latex] 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

stream function

if [latex]{\bf{F}} = {\langle}P, Q{\rangle}[/latex] is a source-free vector field, then stream function g is a function such that [latex]P = g_{y}[/latex], and [latex]Q = -{g_{x}}[/latex]

unit vector field

a vector field in which the magnitude of every vector is [latex]1[/latex]

vector field

measured in [latex]\mathbb{R}^{2}[/latex], an assignment of a vector [latex]{\bf{F}}(x,y)[/latex] to each point [latex](x,y)[/latex] of a subset [latex]D[/latex] of [latex]\mathbb{R}^{2}[/latex]; in [latex]\mathbb{R}^{3}[/latex], an assignment of a vector [latex]{\bf{F}}(x,y,z)[/latex] to each point [latex](x,y,z)[/latex] of a subset [latex]D[/latex] of [latex]\mathbb{R}^{3}[/latex]

vector line integral

the vector line integral of vector field [latex]{\bf{F}}[/latex] along curve [latex]C[/latex] is the integral of the dot product of [latex]{\bf{F}}[/latex] with unit tangent vector [latex]{\bf{T}}[/latex] of [latex]C[/latex] with respect to arc length, [latex]\displaystyle\int_{C} {\bf{F}}\cdot{\bf{T}}ds[/latex]; such an integral is defined in terms of a Riemann sum, similar to a single-variable integral