Chapter 9, Problem 14P

To determine

The graph of vector field $F\left(x,y\right)=\langle y,\frac{1}{x}\rangle$.

Answer to Problem 14P

The graph of vector field $F\left(x,y\right)=\langle y,\frac{1}{x}\rangle$ matches with the graph III.

Explanation of Solution

The vector field is a function of a space which gives a vector quantity at each point. It is a collection of arrows with a given magnitude direction.

The general form of vector field is given below,

$F\left(x,y\right)=xi+yj$ (1)

Compare the vector field $F\left(x,y\right)=\langle y,\frac{1}{x}\rangle$ with equation (1),

$x=y$

$y=\frac{1}{x}$

The vector field at several points shown is below table,

$\left(x,y\right)$	$F\left(x,y\right)$
$\left(1,0\right)$	$\langle 0,1\rangle$
$\left(1,1\right)$	$\langle 1,1\rangle$
$\left(-1,0\right)$	$\langle 0,-1\rangle$
$\left(-1,-1\right)$	$\langle -1,-1\rangle$
$\left(1,2\right)$	$\langle 2,1\rangle$
$\left(-1,2\right)$	$\langle 2,-1\rangle$

From the above table the graph of vector field is given below,

Figure (1)

From Figure (1), it is noticed that all the vectors have different directions and different magnitude. The vectors near to the x-axis have less magnitude than the other vectors.

The above graph is same as the graph III.

Figure (2)

Thus, the graph of vector field $F\left(x,y\right)=\langle y,\frac{1}{x}\rangle$ matches with the graph III.