Chapter 9: Vector Calculus
Section 9.3: Differential Operators
Example 9.3.17
Graph the vector fields F1=y i−x j and F2=F1/x2+y2. Show that ∇×F1=−2 k, but ∇×F2=0, even though the arrows of both fields are tangent to concentric circles, suggesting "rotation" for both.
Solution
Tools≻Load Package: Student Vector Calculus
Set the display format for vectors.
Loading Student:-VectorCalculus
BasisFormatfalse:
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]);
Figure 9.3.17(a) Vector field F1
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]);
Figure 9.3.17(b) Vector field F2
Use the VectorField command to define the vector fields F1 and F2. Be sure to include the third component.
F1≔VectorFieldy,−x,0:F2≔F1/x2+y2:
Compute the curl of each field.
∇×F1 =
simplify∇×F2 =
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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