Chapter 9, Problem 19P

(a)

To determine

To sketch: The streamline start with the point $\left(1,4\right)$.

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 and direction.

The general form of vector field in a plane is given below,

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

The vector which represents the current in a turbulent bay is given below,

$F\left(x,y\right)=\left(x+y\right)i+\left(x-y\right)j$

Compare the vector field $F\left(x,y\right)=\left(x+y\right)i+\left(x-y\right)j$ with equation (1).

$x=x+y$

$y=x-y$

The path followed by the particles in a vector field is called the flow lines or streamlines of the vector field.

The graph of vector field $F\left(x,y\right)=\left(x+y\right)i+\left(x-y\right)j$ with the streamline starting at the point $\left(1,4\right)$ is shown below,

Figure (1)

Thus, the Figure (1) shows the streamline start with the point $\left(1,4\right)$.

(b)

To determine

To sketch: The streamline start with the point $\left(-2,1\right)$.

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 and direction.

The path followed by the particles in a vector field is called the flow lines or streamlines of the vector field.

The graph of vector field $F\left(x,y\right)=\left(x+y\right)i+\left(x-y\right)j$ with the streamline starting at the point $\left(-2,1\right)$ is shown below,

Figure (2)

Thus, the Figure (2) shows the streamline start with the point $\left(-2,1\right)$.

(c)

To determine

To sketch: The streamline start with the point $\left(-1,-2\right)$.

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 and direction.

The path followed by the particles in a vector field is called the flow lines or streamlines of the vector field.

The graph of vector field $F\left(x,y\right)=\left(x+y\right)i+\left(x-y\right)j$ with the streamline starting at the point $\left(-1,-2\right)$ is shown below,

Figure (3)

Thus, the Figure (3) shows the streamline start with the point $\left(-1,-2\right)$.