Chapter 9: Vector Calculus
Section 9.3: Differential Operators
Example 9.3.16
Graph the vector fields F1=x i+y j and F2=F1/x2+y2. Show that ∇·F1=2, but ∇·F2=0, even though the arrows of both fields are radially outward, suggesting "divergence" for both.
Solution
Tools≻Load Package: Student Vector Calculus
Loading Student:-VectorCalculus
Student:-VectorCalculus:-PlotVector(Student:-VectorCalculus:-VectorField(<x,y>),color=red,x=-1..1,y=-1..1,tickmarks=[[-1,1],[-1,1]],arrows=medium,grid=[10,10]);
Figure 9.3.16(a) Vector field F1
Student:-VectorCalculus:-PlotVector(Student:-VectorCalculus:-VectorField(<x,y>/(x^2+y^2)),color=green,x=-1..1,y=-1..1,tickmarks=[[-1,1],[-1,1]],arrows=medium,grid=[10,10]);
Figure 9.3.16(b) Vector field F2
Use the VectorField command to define the vector fields F1 and F2.
F1≔VectorFieldx,y:F2≔F1/x2+y2:
Compute the divergence of each field.
∇·F1 = 2
simplify∇·F2 = 0
The only difference in the fields is the length of the arrows, as seen in Figures 9.3.16(a) and 9.3.16(b). The arrows of each field are directed radially outward. Yet one field has positive divergence everywhere, and the other has no divergence anywhere. The image of a vector field is a global, or "in the large" view, but the divergence is a local, or pointwise, measure. Great caution should be exercised when, from a picture of its arrows, drawing conclusions about the divergence of a field.
Figures 9.3.16(a) and 9.3.16(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 add the option "output=plot." The PlotVector command in the same package will also draw a vector field.
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