Chapter 6 - Vector calculus

Sketch a vector field in two dimensions from a given equation.
Given an equation, be able to identify which of several vector field sketches describe it.
Be able to use CoCalc to produce plots of 2- and 3-dimensional vector fields.
Generate the vector field resulting from the gradient of a scalar function--a conservative field.
Test a vector field to see if it's conservative or not.
Guess a plausible potential function for a conservative field, and test whether it's conservative.
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 $f$.

Parameterize an oriented curve in space, $\myv r(t)$ (from Chapter 9).
Calculate a scalar line integral along an oriented curve.
Calculate a vector line integral along an oriented curve.
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, $\myv F(x,y,z)$.

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