Vector Fields
Vector fields in two dimension
Let and be functions defined on a domain in . A vector field on is a vector-valued function such that for any in ,
A vector field is continuous (or differentiable) on if and are continuous (or differentiable) on .
The applet below illustrates a vector field in 2D space - a vector (arrow) is assigned to every point in the rectangular domain .
You can try the following examples of vector fields:
- (a shear field)
- for (a channel flow)
- (a rotation field)
- , where is a real-valued function. It is called a radial vector field. Of specific interest are the radial vector fields , where is a real number.
Vector fields in three dimension
Let and be functions defined on a domain in . A vector field on is a vector-valued function such that for any in ,
A vector field is continuous (or differentiable) on if and are continuous (or differentiable) on .
The applet below illustrates a vector field in 3D space - a vector (arrow) is assigned to every point in the cuboid .
You can try the following examples of vector fields:
- for
- for (a flow through the cylinder)
- (a rotation field)
- , where (a radial vector field)
Gradient fields
Let be a differentiable function on a domain in . Then
is called a gradient field in 2D space, and is the potential functions for .
Similarly, if is a differentiable function on a domain in , then
is a gradient field in 3D space and is its potential function.
Remark: If is a potential function for a gradient field, then is also a potential function for that gradient field for any constant .
Consider the graph of the potential function . We already know that the gradient field at is always pointing towards the direction of the steepest slope on the graph. Moreover, the gradient field is orthogonal to the level curve of the potential function.
Definition: A vector field (in two or three dimension) is called a conservative field if there exists a function such that
Example: Let be a radial vector field in 3D space, where is a constant. Show that it is a conservative field.
(Note: This vector field is called an inverse-square field. In physics, electric field, gravitational field are all inverse-square fields.)
Answer:
Define .
Then we have
In later chapter, we will learn how to determine whether a vector field is conservative or not.
Exercise: