Computer Algebra Recipes

5.1. CHECKING SOLUTIONS 221
> L:=f->simplify(Laplacian(f,'cartesian'[x,y])):
Applying L to u and to v yields zero, con¯rming that these functions satisfy
Laplace's equation and can be regarded as real potentials.
> L(u); L(v);
0 0
To con¯ne our attention to the region outside the cylinder, two piecewise functions,
pw1 and pw2 , are formed which are equal to zero for x2 + y2 < 1andu
and v, respectively, outside this circular region.
> pw1:=piecewise(x^2+y^2=1,u);
( 2 2 0 x + y < 1
pw1 := x
x + 1 · x2 + y2 x 2 + y 2
> pw2:=piecewise(x^2+y^2=1,v):
A contour plot operator CP is formed to plot the equipotentials for a given
potential function f. The color C must also be speci¯ed. The contours are
drawn for potentials equal to 0:2 n, withn ranging from ¡11 to +11. The grid
spacing is taken to be 90 £ 90, and constrained scaling is imposed.
> CP:=(f,C)->contourplot(f,x=-2..2,y=-2..2,contours=
[seq(0.2*n,n=-11..11)],grid=[90,90],color=C,
scaling=constrained,thickness=2):
The curves corresponding to constant u are colored red, those corresponding
to constant v colored blue. The two sets of curves are superimposed. A blackand-white
version of the plot is shown in Figure 5.4.
> display(fCP(pw1,red),CP(pw2,blue)g);
2
y
–2 x 2
–2
Figure 5.4: Equipotentials and electric ¯eld lines outside a conducting cylinder.