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Tevian Dray
Oregon State University

NCB Deposit  # 64
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Vector Fields
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From the Vector Calculus Bridge Project - Oregon State University

Vector fields are vectors which change from point to point.  A standard example is the velocity of moving air, in other words, wind.  For instance, the current wind pattern in the San Francisco area can be found at < >.  This site has a 2-dimensional representation;  careful reading of the webpage will tell you at what elevation the wind is shown.          How would you represent a vector field in  3  dimensions?  What features are important?  Some simple examples are shown below.  Each can be rotated by clicking and dragging with the mouse.   Explore!

HOLD DOWN the mouse and move it over the vector field images on the left.

The first vector field  
F1  on the left is constant.  It does not change from point to point.  It therefore neither diverges nor spins.

1 = i

One question to ask about a vector field is how it changes from point to point.  Two types of change are especially important:  divergence and curl.  The divergence measures how much  "stuff"  is  "flowing"  away from any given point;  the divergence is a function of position.  The curl measures how much  "stuff"  is rotating around a given point  -- in a given plane;  the curl is itself a vector, and thus can contain information about rotation in all planes.

But the key ideas are  "diverging"  and  "spinning".


2 = x i + y j

The second vector field   F2  on the left is clearly diverging from the center, but not spinning.  Less obvious is that it is also diverging from every other point.

If the wind were blowing like this, ANY two nearby points would find themselves getting further and further apart.  For this reason, the radial vector field of   F2  can be thought of as  "pure divergence". 

= -y i + x j

The third vector field   F3  on the left  is clearly spinning about the center, but does not appear to be diverging.  Less obvious is that it also spins about every other point.

A box placed in this current would not only orbit about the center of the diagram, but also rotate about its own center.  This vector field represents  "pure curl".

= x i + y j  + zk

Now compare  F2  with F4 .   The former is in some sense  2-dimensional, since   F2  is the same in every plane parallel to the  xy-plane, whereas   F4   is 3-dimensional.  Yet both appear similar in the original  2-dimensional representation.

For the next two vector field images be careful to distinguish the  spherical  radial coordinate  r  in  E  from the  cylindrical  radial coordinate  r  in  B.          Physical fields tend to be more complicated than these first four examples.  For instance, the fifth vector field shown is the Coulomb electric field  E  due to a point charge at the origin, while the last is the magnetic field  B  around an infinite wire along the  z- axis carrying a steady current.  It may look as though these fields are again  "pure divergence"  and  "pure curl",  respectively.  However, because these fields are weaker away from the center, some things cancel.  For instance, a small object would not rotate about its center in wind which looked like  B.   It turns out that  E   has divergence only at the origin  (where the divergence is infinite),  and  B  "spins" only along the  z- axis  (where the curl turns out also to be infinite).

Maxwell's equations for electromagnetism predict this behavior  -  the divergence of the electric field tells you where the charges are, and the curl of the magnetic field tells you where the currents are!

Vector equation E

Vector equation B

Please see Dr. Dray's  Bridge Project.
     < >
     < >.
Further properties of the divergence and curl  --  in two dimensions  --  can be discovered using Matthias Kawski's vector field analyzer found at  <  >.
Modern calculus texts will have extensive material on vector calculus. James Stewart,  Calculus, 5th ed., THOMSONBrooks/Cole, 2003,  Chapter 17, pp. 1090-1175.
Raymond Chang,  Physical Chemistry for the Biosciences,  University Science Books, 2005, pp. 25-26.
Donald A. McQuarrie, MATHEMATICAL METHODS for Scientists and Engineers, University Science Books, 2003, (Section 7.1), pp. 191-197.
Michael J. Crowe, A History of Vector Analysis:  The Evolution of the Idea of a Vectorial System, Dover, 1994.
Historical sketch . . .
The definitive history of vector analysis has been written by Michael J. Crowe.  In a nutshell, Hamilton discovered quaternions, Maxwell found equations to describe electromagnetism, and then Gibbs and Heaviside rewrote Maxwell's theory in what is essentially the modern language of vectors.  The  "i, j, k" notation comes from Hamilton's quaternions.
The six vector illustrations shown above were drawn using Maple, then converted to JAVA using  JAVAViewLib.

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James Clerk Maxwell (1831-1879)
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