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.
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 refer to vector-valued functions.
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This first video explains vector fields in detail, with lots of examples and graphs.
video by Dr Chris Tisdell |
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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.
video by Dr Chris Tisdell |
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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.
video by PatrickJMT |
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video by MIP4U |
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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 topic discusses one way to manipulate vector fields, the curl.
Practice
Unless otherwise instructed, plot these vector fields.
\( \vec{F} = 2\hat{i} + 2\hat{j} \)
Problem Statement |
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Unless otherwise instructed, plot the vector field \( \vec{F} = 2\hat{i} + 2\hat{j} \).
Solution |
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video by Dr Chris Tisdell |
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\( \vec{F}(x,y) = -y\hat{i} + x\hat{j} \)
Problem Statement |
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Unless otherwise instructed, plot the vector field \( \vec{F}(x,y) = -y\hat{i} + x\hat{j} \).
Solution |
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video by PatrickJMT |
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Log in to rate this practice problem and to see it's current rating. |
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\( \vec{F}(x,y) = -\hat{i} + \hat{j} \)
Problem Statement |
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Unless otherwise instructed, plot the vector field \( \vec{F}(x,y) = -\hat{i} + \hat{j} \).
Solution |
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video by MIP4U |
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Log in to rate this practice problem and to see it's current rating. |
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\( \vec{F}(x,y) = -x \hat{j} \)
Problem Statement |
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Unless otherwise instructed, plot the vector field \( \vec{F}(x,y) = -x \hat{j} \).
Solution |
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video by MIP4U |
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Log in to rate this practice problem and to see it's current rating. |
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\( \vec{F}(x,y) = x\hat{i} + y\hat{j} \)
Problem Statement |
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Unless otherwise instructed, plot the vector field \( \vec{F}(x,y) = x\hat{i} + y\hat{j} \).
Solution |
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video by MIP4U |
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Log in to rate this practice problem and to see it's current rating. |
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Really UNDERSTAND Calculus
external links you may find helpful |
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The Unit Circle
The Unit Circle [wikipedia]
Basic Trig Identities
Set 1 - basic identities | |||
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\(\displaystyle{ \tan(t) = \frac{\sin(t)}{\cos(t)} }\) |
\(\displaystyle{ \cot(t) = \frac{\cos(t)}{\sin(t)} }\) |
\(\displaystyle{ \sec(t) = \frac{1}{\cos(t)} }\) |
\(\displaystyle{ \csc(t) = \frac{1}{\sin(t)} }\) |
Set 2 - squared identities | ||
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\( \sin^2t + \cos^2t = 1\) |
\( 1 + \tan^2t = \sec^2t\) |
\( 1 + \cot^2t = \csc^2t\) |
Set 3 - double-angle formulas | |
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\( \sin(2t) = 2\sin(t)\cos(t)\) |
\(\displaystyle{ \cos(2t) = \cos^2(t) - \sin^2(t) }\) |
Set 4 - half-angle formulas | |
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\(\displaystyle{ \sin^2(t) = \frac{1-\cos(2t)}{2} }\) |
\(\displaystyle{ \cos^2(t) = \frac{1+\cos(2t)}{2} }\) |
Trig Derivatives
\(\displaystyle{ \frac{d[\sin(t)]}{dt} = \cos(t) }\) |
\(\displaystyle{ \frac{d[\cos(t)]}{dt} = -\sin(t) }\) | |
\(\displaystyle{ \frac{d[\tan(t)]}{dt} = \sec^2(t) }\) |
\(\displaystyle{ \frac{d[\cot(t)]}{dt} = -\csc^2(t) }\) | |
\(\displaystyle{ \frac{d[\sec(t)]}{dt} = \sec(t)\tan(t) }\) |
\(\displaystyle{ \frac{d[\csc(t)]}{dt} = -\csc(t)\cot(t) }\) |
Inverse Trig Derivatives
\(\displaystyle{ \frac{d[\arcsin(t)]}{dt} = \frac{1}{\sqrt{1-t^2}} }\) |
\(\displaystyle{ \frac{d[\arccos(t)]}{dt} = -\frac{1}{\sqrt{1-t^2}} }\) | |
\(\displaystyle{ \frac{d[\arctan(t)]}{dt} = \frac{1}{1+t^2} }\) |
\(\displaystyle{ \frac{d[\arccot(t)]}{dt} = -\frac{1}{1+t^2} }\) | |
\(\displaystyle{ \frac{d[\arcsec(t)]}{dt} = \frac{1}{\abs{t}\sqrt{t^2 -1}} }\) |
\(\displaystyle{ \frac{d[\arccsc(t)]}{dt} = -\frac{1}{\abs{t}\sqrt{t^2 -1}} }\) |
Trig Integrals
\(\int{\sin(x)~dx} = -\cos(x)+C\) |
\(\int{\cos(x)~dx} = \sin(x)+C\) | |
\(\int{\tan(x)~dx} = -\ln\abs{\cos(x)}+C\) |
\(\int{\cot(x)~dx} = \ln\abs{\sin(x)}+C\) | |
\(\int{\sec(x)~dx} = \) \( \ln\abs{\sec(x)+\tan(x)}+C\) |
\(\int{\csc(x)~dx} = \) \( -\ln\abs{\csc(x)+\cot(x)}+C\) |
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Practice Instructions
Unless otherwise instructed, plot these vector fields.