Chapter 15, Problem 1RE

In words, what is a vector field? Give some physical examples of vector fields.

To determine

The definition of vector field and also provide some physical examples of the vector fields.

Answer to Problem 1RE

The vector field is a function in a plane that associates each point in the plane, $P$ . The unique vector, $F\left(P\right)$ , is parallel to the plane.

Explanation of Solution

The vector field is a function in a plane that associates each point in the plane, $P$ . The unique vector, $F\left(P\right)$ , is parallel to the plane.

Similarly, a vector field in the three dimensional space is a function which is associated with each point $P$ in three space a unique vector $F\left(P\right)$ in three space.

The vector field can be expressed as,

$F\left(x,y,z\right)=f\left(x,y,z\right)i+g\left(x,y,z\right)j+h\left(x,y,z\right)k$

Consider the function, $F\left(x,y\right)=\frac{1}{5}\sqrt{y}i$ , which describes the velocity of the current in a system at various depths. The vector field direction is shown below.