Part 4: Flow of a Vector Field (and the JAVA MICROSCOPE!)

If F( x, y, z ) is a vector field, then the flow of F is the set of all curves r(t) that satisfy

dr dt = F,           r(0) = (x,y,z)

(1)

for each point (x,y,z) in 3-dimensional space.  Correspondingly, F( x, y, z )  is called the velocity field of the flow. If F( x, y) = á M(x,y), N(x,y) ñ, then its flow is the set of curves in the xy-plane that satisfy (1) when z = 0. 

 

EXAMPLE 7    Sketch the following family of curves

r(t) = á  x cos(t) - y sin(t),  x sin(t) + y cos(t) ñ

(2)

and then show that it is the flow of the vector field F( x,y) = á -y, x ñ.

Solution: It is easy to show that ||r||²  = x² + y² , thus implying that (2) is a system of concentric circles centered at (0,0) .  Motion on each circle is uniform circular motion, as is shown below along with the vector field. 

Moreover, if X = x cos(t) - y sin(t), and Y = x sin(t) + y cos(t), then

dr dt = á  -x sin(t)  -  y cos(t),  x cos(t) - y sin(t) ñ = á -Y,  X ñ

Specifically, r' (0) = á -y, x ñ,  which implies that  F( x,y) = á -y, x ñ .        

In particular, the flow in example 7 is rotation about the origin.  Moreover, the curl of the vector field F( x,y) = á -y, x ñ in example 7 is 

curl( F) =  0,0,  ¶ ¶x ( x) -  ¶ ¶y ( -y) = á 0,0,2 ñ

That is, F( x,y) = á -y, x ñ has a constant curl that is parallel to the axis of rotation (i.e., the z-axis), and it can be shown in general that the curl of a vector field is a measure of the amount of local rotation in the flow (i.e., rotation near a given point). Indeed, because gradients of a function U(x,y) are normal to level curves of U(x,y), they are practically parallel to each other locally.  Thus, there is no local rotation, thus implying that the curl of a conservative field is zero.  

Similarly, the divergence of a vector field  is a measure of the amount of local expansion of the flow--i.e., how much the area of a region increases or decreases as the region moves through the flow.  The divergence of the vector field F( x,y) = á -y, x ñ in example 7 is zero because rotation of a region does not change its area..  

Perhaps these ideas are better understood if we explore them directly.  To do so, we use the Java Microscope developed by Shannon Holland and Dr. Matthias Kawski.   Enter a vector field by entering functions for M and N, and then click on "Draw Vector Field".  Click on "Flows", then click and drag to create a rectangle.  Releasing it will show you how the rectangle "flows" within the vector field.  If its sides begin to rotate, then the curl is non-zero. If the rectangle begins to expand, then the divergence is non-zero.

Java Microscope

It may take some time for the graph to load after pressing the "Draw Vector Field" button. To create a magnifying lens, drag your mouse across the vector field. Use numpoints less than 20 (20 X 20 vector lines) for quick response. Larger values will work, but take much longer. For more information on the Java Microscope read , An interactive JAVA microscope to visualize divergence and curl via zooming.  NEW:   be sure to check out the new and improved flow routines.

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