vektorfeld_rotation_divergenz.sagews

# Zum Beispiel in der Vorlesung (Folie 58/59)
var ('x y') # Variablen deklarieren
F1=vector([x,-y]) # Funktion definieren

view(F1) # Funktion symbolisch darstellen lassen (als Zeilenvektor)
V1=plot_vector_field(F1,(x,-3,3),(y,-3,3), color='blue') # Vektorfeld Plotobjekt erstellen

show(V1,gridlines=True, aspect_ratio=1,axes_labels=['$x$','$y$'])# Vektorfeld plotten

(x, y)

$\displaystyle \left(x,\,-y\right)$

var('x y z') # Jetzt Definition in 3D, zur Berechnung der Rotation und Divergenz
F=vector([x,-y, 0])
view(F)
V=plot_vector_field3d(F,(x,-2,2),(y,-2,2),(z,-2,2), colors=['red','green','blue'])
show(V) # vektorfeld plotten
def divergence(F):
    assert(len(F) == 3) # Test ob Vektor in 3D gegeben ist
    return diff(F[0],x) + diff(F[1],y) + diff(F[2],z)
def curl(F):
    assert(len(F) == 3)
    return vector([diff(F[2],y)-diff(F[1],z), diff(F[0],z)-diff(F[2],x), diff(F[1],x)-diff(F[0],y)])

curl(F) # Rotation
divergence(F) # Divergenz

(x, y, z)

$\displaystyle \left(x,\,-y,\,0\right)$

3D rendering not yet implemented

(0, 0, 0) 0

3D rendering not yet implemented

(0, 0, 0) 0

var('x y z') # Beispiel aus der Vorlesung
#F=vector([-y/sqrt(x^2+y^2),-x/sqrt(x^2+y^2),0])
F=vector([x/(sqrt(x^2+y^2))^3,y/(sqrt(x^2+y^2))^3,z/(sqrt(x^2+y^2))^3])
view(F)

#V=plot_vector_field3d(F,(x,-2,2),(y,-2,2),(z,-2,2), colors=['red','green','blue'])
#show(V) # vektorfeld plotten
rot=curl(F)
div=divergence(F)
show(rot)
show(div)

(x, y, z)

$\displaystyle \left(-\frac{3 \, y z}{{\left(x^{2} + y^{2}\right)}^{\frac{5}{2}}},\,\frac{3 \, x z}{{\left(x^{2} + y^{2}\right)}^{\frac{5}{2}}},\,0\right)$

$\displaystyle -\frac{3 \, x^{2}}{{\left(x^{2} + y^{2}\right)}^{\frac{5}{2}}} - \frac{3 \, y^{2}}{{\left(x^{2} + y^{2}\right)}^{\frac{5}{2}}} + \frac{3}{{\left(x^{2} + y^{2}\right)}^{\frac{3}{2}}}$

var('x y z') # Zur Aufgabe 8, Übung 2
F=vector([x*y^2,2*x^2*y*z, -3*y*z])
view(F)
V=plot_vector_field3d(F,(x,-2,2),(y,-2,2),(z,-2,2), colors=['red','green','blue'])
show(V) # vektorfeld plotten
rot=curl(F)

show(rot)

(x, y, z)

$\displaystyle \left(x y^{2},\,2 \, x^{2} y z,\,-3 \, y z\right)$

3D rendering not yet implemented

$\displaystyle \left(-2 \, x^{2} y - 3 \, z,\,0,\,4 \, x y z - 2 \, x y\right)$