Vector Fields and the vector operations divergence and curl.

Purpose

The purpose of this lab is to acquaint you with some Maple commands for visualizing and computing the divergence and curl of vector fields.

Background

A vector field is a vector-valued function that associates a vector with each point in its domain. For example the function produces a vector at each point . Probably the most familiar example of a vector field is the velocity field in a fluid in motion. Here you would have a velocity vector that describes the velocity of the fluid at a point . If the flow is steady, then will not depend on t, but it still could depend on the position in space. Another example that we have seen already is the gradient of a scalar-valued function . The electric field and the magnetic field might also be familiar to you.

Just as we did with constant vectors, we can represent vector fields by ordered pairs (in two dimensions) or triples (in three dimensions) of component functions. For example, we could also represent our first example by . The dimension of a vector field is the number of component functions it has. Thus, our example is a two-dimensional vector field and the fluid velocity is a three-dimensional vector field. Note that the dimension is not always the same as the number of independent variables.

Two-dimensional vector fields are often represented graphically by what are called vector or arrow plots. To see how these are generated, suppose you have a vector field and you fix a point . To represent the vector field at this point you draw a vector with its tail at in the direction of , with the length of the vector being some scaling factor times the length of . You then repeat this process at a grid of points. If you think of a vector field as representing the flow of a fluid, then an arrow plot gives you an idea of the paths fluid particles will take.

Examples are probably easier to understand than explanations, and Maple has a procedure that draws arrow plots for two-dimensional vector fields. The procedure is called fieldplot and the following example shows how to use it. Note that you have to load the plots package before you can use it.

  > with(plots):

  > fieldplot([-x,y],x=-4..4,y=-4..4);

  > fieldplot([cos(x+y),-sin(y)],x=-6..6,y=-6..6,grid=[25,25]);

The last line of the examples shows how to use the grid option to change the number of grid points. See the help for fieldplot for more details. There is also a fieldplot3d command for producing arrow plots of three dimensional vector fields, but we will not make use of it.

We have already seen the symbol in the definition of the gradient, but there are two other calculus operations on vector fields involving it. To understand them, it is perhaps best to think of in three dimensions as the operator

with the obvious change in definition for in two dimensions. Then the gradient of a function is obtained by ``scalar multiplication'' of f on the left by the vector operator .

Whereas the gradient applies to a scalar function, the divergence and curl are applied to vector-valued functions. Suppose is a vector-valued function with component functions , , and . That is, we can write

Then the divergence of the vector field , written is the scalar function

The divergence is also often written as , which is useful for remembering the formula for the divergence: think of it as the dot product of the operator and the vector function . For example, if

then

Having seen the dot product reappear, is it any surprise that the cross product is waiting in the wings? The curl of a three-dimensional vector field sometimes written , is more often represented as . As you might expect, the curl of a vector field produces another vector field. The formula for is

The formula is best remembered as the symbolic determinant

The linalg package in Maple has commands grad, diverge, and curl that perform these vector operations. Examples are shown below. The last example shows how to work with an arbitrary vector function .

  > with(linalg):

Warning: new definition for   norm

Warning: new definition for   trace

  > f := vector([x,y^3,x*y*z]);

  > diverge(f,[x,y,z]);

  > curl(f,[x,y,z]);

  > F:= (x,y,z) -> vector([P(x,y,z),Q(x,y,z),R(x,y,z)]) ;

  > diverge(F(x,y,z),[x,y,z]);

The Maple procedure diverge will work on vector fields of any dimension, but curl expects a three-dimensional vector field. The list [x,y,z] is used to specify the independent variables in their natural order.

Exercises

1. Plot the following vector fields with the fieldplot command.
   1. .
   2. .
   3. .
   4. .
2. For this problem think of the vector field as representing fluid flow. Can you find a two dimensional vector fields that satisfy the following conditions?
   1. All the fluid is flowing away from the center.
   2. All the fluid is flowing toward the center.
   3. The fluid is going around in the clockwise direction.
   4. Fluid in the right half plane is going to the right. Fluid in the left half plane is going to the left.
3. Using Maple, or otherwise, compute the divergence and the curl of the following vector fields.
   1. .
   2. .
   3. .
4. Show that for any vector field .
5. If is a scalar function and is a three-dimensional vector, show that the following identities are true.
   1. .
   2.