Summary of 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].

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]

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

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

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]