Vector Fields
Textbook Reference
| Primary source | OpenStax Calculus Volume 3, Section 6.1: “Vector Fields” |
| Direct link | https://openstax.org/books/calculus-volume-3/pages/6-1-vector-fields |
| Textbook used in class | Stewart, Calculus, Section 16.1: “Vector Fields” (Examples 1, 2, 3) |
Quick Reference
Vector field in $\mathbb{R}^2$: $\mathbf{F}(x,y) = P(x,y)\,\mathbf{i} + Q(x,y)\,\mathbf{j}$.
Vector field in $\mathbb{R}^3$: $\mathbf{F}(x,y,z) = P\,\mathbf{i} + Q\,\mathbf{j} + R\,\mathbf{k}$.
A vector field assigns a vector to each point in a region. Sketching it means drawing the vector $\mathbf{F}(x,y)$ with its tail at $(x,y)$ for a grid of representative points.
Motivation
A vector field models any physical quantity that has both magnitude and direction at every point in space: the velocity of wind at each location, the force of gravity at each point near Earth, or the electric field around a charge. Vector calculus -- line integrals, surface integrals, curl, divergence, and the major theorems -- is the language for working with these fields.
Key Concepts
1. Sketching Vector Fields
To sketch $\mathbf{F}(x,y)$, evaluate $\mathbf{F}$ at several representative points and draw the resulting vector (arrow) at each point. The arrows reveal the structure of the field: does it rotate? converge? diverge?
Example: $\mathbf{F}(x,y) = -y\,\mathbf{i} + x\,\mathbf{j}$. At $(1,0)$: $\mathbf{F} = \langle 0,1\rangle$ (points up). At $(0,1)$: $\mathbf{F} = \langle -1,0\rangle$ (points left). The field rotates counterclockwise around the origin.
2. Common Physical Fields
- Gravitational field: $\mathbf{F} = -GM/(|\mathbf{r}|^3)\,\mathbf{r}$ (pointing toward origin, decreasing with distance).
- Constant field: $\mathbf{F} = \langle a, b\rangle$ (same vector everywhere -- uniform wind or gravity near Earth’s surface).
- Radial field: $\mathbf{F} = \mathbf{r}/|\mathbf{r}|$ (unit vector pointing away from origin).
Worked Example
Sketch the vector field $\mathbf{F}(x,y) = x\,\mathbf{i} + y\,\mathbf{j}$. (Stewart 16.1, Example 1.)
At $(1,0)$: $\mathbf{F} = \langle 1,0\rangle$. At $(0,1)$: $\mathbf{F} = \langle 0,1\rangle$. At $(1,1)$: $\mathbf{F} = \langle 1,1\rangle$. At $(-1,-1)$: $\mathbf{F} = \langle -1,-1\rangle$.
All vectors point directly away from the origin. The field is a radial expansion. Note $\mathbf{F}(x,y) = \langle x,y\rangle = \mathbf{r}$: this is the position vector field.
the magnitude of $\mathbf{F}$ at a point can be read from the length of the arrow in a field sketch. In practice, field sketches are often drawn with arrows scaled or normalized for readability (otherwise, large vectors would dominate and small vectors would be invisible). The length convention varies by textbook. When interpreting a sketch, check whether the arrows are scaled to actual magnitude or drawn at unit length with direction only.
Common Misconceptions
the arrows in a vector field diagram show the path that a particle follows through space.
This is the action-view-of-function error. Each arrow in a vector field diagram represents the vector value of $\mathbf{F}$ at the base point of the arrow, not a trajectory. A particle released at a point moves tangent to the field at each instant; the resulting path (a flow line or streamline) is a curve, not a single arrow. The arrow at $(x_0, y_0)$ shows the instantaneous direction and relative magnitude of the field there, nothing more.
Leveled Practice
Problem 1. Describe the vector field $\mathbf{F}(x,y) = \mathbf{i}$ (constant field $\langle 1,0\rangle$ everywhere). What does it represent physically?
Show answer
At every point, $\mathbf{F} = \langle 1,0\rangle$: a horizontal arrow pointing right. This represents a uniform wind or constant horizontal force (like a uniform electric field).