Vector fields and Line integrals, Part 6

The graphical and symbolic representations for four different vector fields are shown below. Match the graphical and symbolic representations.

$\begin{displaymath}h_{1}(x,y,z)=\left[ \begin{array}{c} x \\ y \\ z \\ \end{array} \right] \end{displaymath}$

$\begin{displaymath}h_{2}(x,y,z)=\left[ \begin{array}{c} x^{2} + y^{2} \\ x^{2} - y^{2} \\ 0 \\ \end{array} \right] \end{displaymath}$

$\begin{displaymath}h_{3}(x,y,z)=\left[ \begin{array}{c} -y \\ x \\ 0 \\ \end{array} \right] \end{displaymath}$

$\begin{displaymath}h_{4}(x,y,z)=\left[ \begin{array}{c} \sin (\pi x) \\ 0 \\ 1 \\ \end{array} \right] \end{displaymath}$
Explain how vectors and dot products can be used to calculate the work done when a force $\vec{F}$ moves an object along a path described by a position vector $\vec{s}$ . Under what circumstances can movement take place without doing any work?

Consider the two dimensional vector field with the symbolic representation $\begin{displaymath}\vec{v}(x,y)=\left[ \begin{array}{c} \frac{x}{\sqrt{x^2 + y^2}} \\ \frac{y}{\sqrt{x^2 + y^2}} \\ \end{array} \right] \; . \end{displaymath}$ Describe (in words) a closed curve that will give positive work if $\vec{v}(x,y)$ represents the force acting and your curve represents the path that the object moves along. Do the same for negative work.

Summarize what you have learned about the relationship between the curve, the vector field and the sign of the work done. Give plausibility arguments for your conclusions, using what you have learned about approximating work by use of collections of straight line segments and sums.

Describe a procedure for calculating the work done by a force $\vec{F}(x,y)$ acting on an object that moves along a path parameterized by $\gamma (t) = (\gamma_{1} (t),\gamma_{2} (t))$ , with $a \leq t \leq b$ .

	$\begin{displaymath}h_{1}(x,y,z)=\left[ \begin{array}{c} x \\ y \\ z \\ \end{array} \right] \end{displaymath}$
	$\begin{displaymath}h_{2}(x,y,z)=\left[ \begin{array}{c} x^{2} + y^{2} \\ x^{2} - y^{2} \\ 0 \\ \end{array} \right] \end{displaymath}$
	$\begin{displaymath}h_{3}(x,y,z)=\left[ \begin{array}{c} -y \\ x \\ 0 \\ \end{array} \right] \end{displaymath}$
	$\begin{displaymath}h_{4}(x,y,z)=\left[ \begin{array}{c} \sin (\pi x) \\ 0 \\ 1 \\ \end{array} \right] \end{displaymath}$