Bilder zu Divergenz und Rotation

Buch:Höhere Mathematik sehen und verstehen, Haftendorn, Riebesehl, Dammer, 
Springer Spektrum, Feb. 2021
 Datei Bilder Div Curl.nb zu Abschnitt 3.2.5 Seite 244, Abb. 3.16, 3.20 und 3.22

Div = Rot = 0

F={x,-y};

d0r0=VectorPlot[F,{x,-2,2},{y,-2,2},VectorPoints->10]

Div[F,{x,y}]

Curl[F,{x,y}]

Div ≠0, Rot = 0

F={x,y};

d2r0=VectorPlot[F,{x,-2,2},{y,-2,2},VectorPoints->10]

Div[F,{x,y}]

Curl[F,{x,y}]

Div = 0, Rot ≠ 0

F={-y,x};

d0r2=VectorPlot[F,{x,-2,2},{y,-2,2}]

Div[F,{x,y}]

Curl[F,{x,y}]

Div ≠ 0, Rot ≠ 0

F={-y+x,x+y};

d2r2=VectorPlot[F,{x,-2,2},{y,-2,2}]

Div[F,{x,y}]

Curl[F,{x,y}]

ColorData["Gradients"];

StreamDensityPlot[F,{x,-2,2},{y,-2,2},ColorFunction->"LightTemperatureMap",Frame->False,StreamScale->Medium]

Div wechselt von negativ auf positiv

F={1+(x^2),0};

VectorPlot[F,{x,-2,2},{y,-2,2}]

Div[F,{x,y}]

Curl[F,{x,y}]

ColorData["Gradients"];

StreamDensityPlot[F,{x,-2,2},{y,-2,2},ColorFunction->"LightTemperatureMap",Frame->False,StreamScale->Medium]

Div wechselt von negativ auf positiv, interessanter

F={(x^2)+1,y x};

VectorPlot[F,{x,-2,2},{y,-2,2}]

Div[F,{x,y}]

Curl[F,{x,y}]

ColorData["Gradients"];

StreamDensityPlot[F,{x,-2,2},{y,-2,2},ColorFunction->"LightTemperatureMap",Frame->False,StreamScale->Medium]

Curl wechselt von negativ auf positiv

F={0,1+(x^2)};

VectorPlot[F,{x,-2,2},{y,-2,2}]

Div[F,{x,y}]

Curl[F,{x,y}]

ColorData["Gradients"];

StreamDensityPlot[F,{x,-2,2},{y,-2,2},ColorFunction->"LightTemperatureMap",Frame->False,StreamScale->Medium]

Curl wechselt von negativ auf positiv, interessanter

F={-y x,(x^2)+1};

VectorPlot[F,{x,-2,2},{y,-2,2}]

Div[F,{x,y}]

Curl[F,{x,y}]

ColorData["Gradients"];

StreamDensityPlot[F,{x,-2,2},{y,-2,2},ColorFunction->"LightTemperatureMap",Frame->False,StreamScale->Medium]

Sieht aus, als wäre Rotation vorhanden, ist es aber nicht!

Grad[(1/2)Log[(x^2)+(y^2)],{x,y}]

F={x,y}(1/((x^2)+(y^2)));

VectorPlot[F,{x,-2,2},{y,-2,2}]

Div[F,{x,y}]//Simplify

Curl[F,{x,y}]

ColorData["Gradients"];

StreamDensityPlot[F,{x,-2,2},{y,-2,2},ColorFunction->"LightTemperatureMap",Frame->False,StreamScale->Medium]

Komplizierteres mit Div ≠ 0, Rot ≠ 0

F={Cos[x+y^3],Sin[y+x^3]};

F={Cos[x+y^3],Sin[y^2+2x]};

F={Cos[x^2+2y],Sin[y^2+2x]};

F[x,y]//TeXForm

VectorPlot[F,{x,-2,2},{y,-2,2},BaseStyle->FontSize->20]

StreamDensityPlot[{F,Norm[F[x,y]]}//Evaluate,{x,-2,2},{y,-2,2},ColorFunction->"RedGreenSplit",Frame->False,StreamPoints->Fine]

Div[F,{x,y}]

Curl[F,{x,y}]

Partition[ColorData["Gradients"],5]//MatrixForm

Grid[Partition[Show[ColorData[#,"Image"],ImageSize->110]&/@ColorData["Gradients"],5],Spacings->.5]

DensityPlot[x,{x,-1,1},{y,0,0.1},ColorFunction->"TemperatureMap",AspectRatio->0.1]

DensityPlot[x,{x,-1,1},{y,0,0.1},ColorFunction->"RedGreenSplit",AspectRatio->0.1]

DensityPlot[x,{x,-1,1},{y,0,0.1},ColorFunction->"GreenPinkTones",AspectRatio->0.1]

DensityPlot[x,{x,-1,1},{y,0,0.1},ColorFunction->"BrightBands",AspectRatio->0.1]

divplot=StreamDensityPlot[{F,Div[F,{x,y}]}//Evaluate,{x,-2,2},{y,-2,2},ColorFunction->"GreenPinkTones",Frame->False,StreamPoints->Fine]

curlplot=StreamDensityPlot[{F,Curl[F,{x,y}]}//Evaluate,{x,-2,2},{y,-2,2},ColorFunction->"TemperatureMap",Frame->False,StreamPoints->Fine]