Vector fields and Line integrals, Part 1

Vector Fields and Line integrals

Part 1: Vector Valued Functions

You are probably familiar with the notion of a function, especially as an operation that takes some input (e.g., one or more numerical values) and returns an output (often a single numerical value). Here we examine functions that take one or more numerical inputs and produce a vector as an output.

Evaluate the function defined by $\begin{displaymath}f(x,y)=\left[ \begin{array}{c} x + y \\ x - y \\ \end{array} \right] \; . \end{displaymath}$ at each of the points
- (x, y) = (1,1)
- (x, y) = (1,0)
- (x, y) = (0,-1)
- (x, y) = (-1,1)
Find the graphical representation of the vector at each of these points.

The diagrams below are representations of the vector-valued functions (vector fields)

$\begin{displaymath}f(x,y)=\left[ \begin{array}{c} x + y \\ x - y \\ \end{array} \right] \end{displaymath}$
and
$\begin{displaymath}g(x,y,z)=\left[ \begin{array}{c} 2x \\ 2y \\ 1 \\ \end{array} \right] \; . \end{displaymath}$

(There are commands in your computer algebra system worksheet to produce similar diagrams that you can manipulate and experiment with.) Describe in your own words what these diagrams are actually showing you.
The table below shows graphical representations A, B, C, D, E for several vector fields, as well as symbolic formulas for the same fields but not in the same order. Match the graphical and symbolic representations. How can you use your computer algebra system to simplify this matching problem?

	$\begin{displaymath}g_{1}(x,y)=\left[ \begin{array}{c} \frac{x}{y} \\ \frac{y}{x} \\ \end{array} \right] \end{displaymath}$
	$\begin{displaymath}g_{2}(x,y,z)=\left[ \begin{array}{c} y \\ x \\ z \\ \end{array} \right] \end{displaymath}$
	$\begin{displaymath}g_{3}(x,y)=\left[ \begin{array}{c} \sqrt{x^{2} + y^{2}} \\ -\sqrt{x^{2} + y^{2}} \\ \end{array} \right] \end{displaymath}$
	$\begin{displaymath}g_{4}(x,y,z)=\left[ \begin{array}{c} \frac{y}{z} \\ \frac{x}{z} \\ z \\ \end{array} \right] \end{displaymath}$
	$\begin{displaymath}g_{5}(x,y)=\left[ \begin{array}{c} \sin (y) \\ x \\ \end{array} \right] \end{displaymath}$

The diagram below shows a graphical representation of a three dimensional vector field, . Experiment with your computer algebra system to find a possible symbolic representation for . Explain how you can check whether your symbolic representation is close to or not.

modules at math.duke.edu