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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 VectorValued 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. Vectorvalued 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.

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. 
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 