Summary of Vector Fields

A vector field assigns a vector ${\bf{F}}(x,y)$ to each point $(x,y)$ in a subset $D$ of $\mathbb{R}^{2}$ or $\mathbb{R}^{3}$ . ${\bf{F}}(x,y,z)$ to each point $(x,y,z)$ in a subset $D$ of $\mathbb{R}^{3}$ .
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 ${\bf{F}}$ is called conservative if there exists a scalar function $f$ such that $\nabla{f}={\bf{F}}$ .

Key Equations

Vector field in $\mathbb{R}^{2}$
${\bf{F}}(x,y)=\langle{P(x,y),Q(x,y)}\rangle$ or ${\bf{F}}(x,y)=P(x,y){\bf{i}},Q(x,y){\bf{j}}$
Vector field in $\mathbb{R}^{3}$
${\bf{F}}(x,y,z)=\langle{P(x,y,z),Q(x,y,z),R(x,y,z)}\rangle$ or ${\bf{F}}(x,y,z)=P(x,y,z){\bf{i}},Q(x,y,z){\bf{j}},R(x,y,z){\bf{k}}$

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

gradient field: a vector field ${\bf{F}}$ for which there exists a scalar function $f$ such that $\nabla{f}={\bf{F}}$ in other words, a vector field that is the gradient of a function; such vector fields are also called conservative

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 $(x,y)$ 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

unit vector field: a vector field in which the magnitude of every vector is $1$

vector field: measured in $\mathbb{R}^{2}$ , an assignment of a vector ${\bf{F}}(x,y)$ to each point of a subset of $\mathbb{R}^{2}$ ; in $\mathbb{R}^{3}$ , an assignment of a vector ${\bf{F}}(x,y,z)$ to each point of a subset of $\mathbb{R}^{3}$