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 < http://sfports.wr.usgs.gov/cgi-bin/wind/windbin.cgi >. 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.
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F1 = i The first vector field F1 on the left is constant. It does not change from point to point. It therefore neither diverges nor spins. 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". |
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F2 = x i + y j The second vector field F1 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". |
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F3 = - 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". |
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F4 = x i + y j + z k 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. |
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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! |
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