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Chapter 6 - VECTOR CALCULUS - Learning objectives
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<h1>Chapter 6 - Vector calculus</h1>
<h4>Learning Objectives</h4>

<h3>Vector Fields</h3>
<ul>
<li>Sketch a vector field in two dimensions from a given equation.
<li>Given an equation, be able to identify which of several vector field sketches describe it.
<li>Be able to use CoCalc to produce plots of 2- and 3-dimensional vector fields.
<li>Generate the vector field resulting from the gradient of a scalar function--a <b>conservative</b> field.
<li>Test a vector field to see if it's conservative or not.
<li>Guess a plausible potential function for a conservative field, and test whether it's conservative.
<li>Plot the contour plot of a potential function $f(x,y)$, and visually verify whether a particular vector field, $\myv F$, could be the gradient of $f$. 
</ul>

<h3>Line integrals</h3>

<ul>
<li>Parameterize an oriented curve in space, $\myv r(t)$ (from Chapter 9).
<li>Calculate a scalar line integral along an oriented curve.
<li>Calculate a vector line integral along an oriented curve.
<li>Use a line integral to compute the work done, $\int \myv F\cdot \,d\myv s$  in moving an object along a curve in a vector force field, $\myv F(x,y,z)$.
</ul>


<h3>Fundamental theorems &amp; Conservative vector fields</h3>
<ul>
<li>Find, and explain how to find, a potential function for a conservative vector field.
<li>Use the Fundamental Theorem for Line Integrals (the 'Cosmic shortcut') to evaluate a line integral in a vector field.
<li>Explain the interconnected properties of conservative vector fields:
$$\myv \grad \times \myv F=0 \ \Leftrightarrow \ \myv F=\myv \grad f\  \Leftrightarrow \ 
\oint \myv F\cdot d\myv s = 0.$$
<li>Carry out the Clairaut / curl ($\myv \grad \times$) test on a 2- or 3-d vector field to determine whether it is conservative.
<li>Use the paddle-wheel test to qualitatively estimate the curl at any point in a 2-d plot of a vector field.
<li>Identify simple vs closed curves; Identify connected and simple connected regions.
<li>Use Green's theorem to evaluate a line integral by means of a double-integral, or vice-versa.
</ul>



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