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Chapter 9: Vector Calculus
Section 9.3: Differential Operators
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Example 9.3.17
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Graph the vector fields and . Show that , but , even though the arrows of both fields are tangent to concentric circles, suggesting "rotation" for both.
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Solution
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Tools≻Load Package: Student Vector Calculus
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Set the display format for vectors.
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Loading Student:-VectorCalculus
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Student:-VectorCalculus:-PlotVector(Student:-VectorCalculus:-VectorField(<y,-x>),color=red,x=-1..1,y=-1..1,tickmarks=[[-1,1],[-1,1]],arrows=medium,grid=[10,10]);
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Figure 9.3.17(a) Vector field
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Student:-VectorCalculus:-PlotVector(Student:-VectorCalculus:-VectorField(<y,-x>/(x^2+y^2)),color=green,x=-1..1,y=-1..1,tickmarks=[[-1,1],[-1,1]],arrows=medium,grid=[10,10]);
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Figure 9.3.17(b) Vector field
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Use the VectorField command to define the vector fields and .
Be sure to include the third component.
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Compute the curl of each field.
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=
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=
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The only difference in the fields is the length of the arrows, as seen in Figures 9.3.17(a) and 9.3.17(b). The arrows of each field are tangent to concentric circles. Yet one field has nonzero curl, and the other has zero curl. The image of a vector field is a global, or "in the large" view, but the curl is a local, or pointwise, measure. Great caution should be exercised when, from a picture of its arrows, drawing conclusions about the curl of a field.
Figures 9.3.17(a) and 9.3.17(b) can be drawn with the fieldplot command in the plots package, or more easily with the VectorField command itself in the Student VectorCalculus package. Simply pass the option "output=plot." The PlotVector command in the same package will also draw a vector field.
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