<?php $classpath="/213/"; include "../../classes/1.h"; ?>
<title>
Vector Fields [6.1]
</title>
<meta name="week" content="11">
<!--meta name="nolink" content="nolink"-->
<?php include "../../classes/2.h" ?>

<h1>
Vector fields
</h1>
<p class="show"><img src="../_g/pidgy.jpg">
<p><a href="https://openstax.org/books/calculus-volume-3/pages/1-introduction"><img src="../amain/calc3.png" style="width:160px;" class="leftalign">
</a>
From now on, we'll be using Gilbert Strang's, Calculus Vol 3 textbook.  Chapter 6 is the beginning of his Vector Calculus section.


<h3>2D vector field</h3>
<p class="show"><img src="0g/13.1.def1.jpg"><br>
<img src="0g/13.1.arrows.jpg">

<h3>Examples</h3>
<p class="show">$$\myv F(x,y)=-y\uv i +x \uv j$$
<br><img src="0g/13.1.vf5.jpg">

<h3>Examples</h3>
<p class="show"><img src="0g/13.1.vf1.jpg">

<h3>Examples</h3>
<p class="show"><img src="0g/13.1.vf2.jpg">

<h3>To do /mvhandouts/</h3>
<ul class="else todo"><li>06.1.VectorFieldSketches - 
Vector Field Sketches by hand
</ul>

<h3>3D vector field</h3>
<p class="show"><img src="0g/13.1.def2.jpg"><br>
<img src="0g/13.1.arrows3d.jpg">


<h3>Examples</h3>
<p class="show"><img src="0g/13.1.ex3d.jpg">


<h3>Examples</h3>
<p class="show">$$\myv F(x,y,z)=z\uv k$$
<br><img src="0g/13.1.vf6.jpg">



<div class="justme">
<h3>Mathematica</h3>
<p class="show">Download and open  "<b>13.1.MticaVectorFields.nb</b>"
in the handouts folder...

<ul><li><tt>VectorPlot</tt> - 2D vector fields<br>
$$\myv F(x,y)=y\uv i -x\uv j$$<br>
<img src="0g/13.1.mtica1.jpg"><br>

<li><tt>VectorPlot3D</tt> - 3D vector fields<br>
$$\myv F(x,y)=y\uv i+z\uv j+x\uv k$$
<img src="0g/13.1.mtica2.jpg"><br>
</ul>


<p><img src="0g/13.1.scalem1.jpg" class="big"><br>
Use  <tt>VectorScale $\to$ Automatic</tt> if necessary.  
</div>

<h3>To Do</h3>
<p>See the Jupyter notebook "vector fields" in your <b>handouts</b> folder <b>on cocalc.com</b> for examples of how to make plots of vector fields in CoCalc.

<p class="show">Sketch these in CoCalc
<ul class="todo">
<li>$\myv F(x,y)=x^2 \uv i +x^3 \uv j$
<li>$\myv F(x,y)=y^3 \uv i +y^2 \uv j
$
<li>$\myv F(x,y)=(x+y) \uv i +(x-y) \uv j$<br><br>

<li>$\myv F(x,y,z)=y \uv i + z \uv j + x \uv k$
<li>$\myv F(x,y,z)=y\uv i -2 \uv j +x \uv k$
<li>$\myv F(x,y,z)=1\uv i -y^2 \uv j +z\uv k$
</ul>

<p><img src="0g/13.1.scalem1.jpg" class="big"><br>


<p>
<img src="0g/13.1.scaleagain.jpg" class="big"><br>












<h4 class="handout">Image credits</h4>
<p class="handout"><a href=""></a>

<?php include "../../classes/3.h" ?>