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17calculus > vector fields

Vector Fields

On this page we will give you an introduction to vector fields and how to draw them. We also have a few practice problems. Two main ways to work with vector fields involve the divergence and the curl.

Difference Between Vector Functions, Vector Fields and Vector-Valued Functions

These three terms are easily confused and some books and instructors interchange them. In general, vector functions are parametric equations described as vectors. Vector fields usually define a vector to each point in the plane or in space to describe something like fluid flow, air flow and similar phenomenon. Vector-valued functions may refer to either vector functions or vector fields. Look carefully at the context and check with your instructor to make sure you understand what they are talking about.

In all three cases, you need to look at the context to see what is being discussed. To avoid confusion, we do not use the term vector-valued function on this site but some of the instructors in the videos we use, do refer to vector-valued functions.

This first video explains vector fields in detail, with lots of examples and graphs.

Dr Chris Tisdell - Intro to vector fields

As explained in the last video, you have already seen vector fields if you have learned how to calculate gradients since the gradient assigns a vector to each point in space.

Here is a second video explaining vector fields. He goes into more detail about applications and why it is important to have a correct understanding of them. There is some repetition but it is important to think about this from different angles to get a good perspective.

Dr Chris Tisdell - What is a vector field?

The videos above should be enough to explain the basics of vector fields. If you would like a couple of other perspectives, here are two more video clips explaining the same concepts.

PatrickJMT - Vector Fields

MIP4U - Vector Fields

Okay, so that should be enough explanation to get you started on vector fields. You can find a few practice problems below.
After that, the next topics discuss ways to manipulate vector fields, called the divergence and the curl.

next: divergence →
next: curl →

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Practice Problems

Instructions - - Unless otherwise instructed, plot these vector fields.

Level A - Basic

Practice A01

\( \vec{F} = 2\hat{i} + 2\hat{j} \)

solution

Practice A02

\( \vec{F}(x,y) = -y\hat{i}+x\hat{j} \)

solution

Practice A03

\( \vec{F}(x,y) = -\hat{i}+\hat{j} \)

solution

Practice A04

\( \vec{F}(x,y) = -x\hat{j} \)

solution

Practice A05

\( \vec{F}(x,y) = x\hat{i} + y\hat{j} \)

solution

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