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A brief discussions on derivative and the"Gradient"

H.Tahsiri

Vector Gradient

H.Tahsiri

<<Calculus`VectorAnalysis`
<<Graphics`PlotField3D`
<<Graphics`PlotField`
<<Graphics`Graphics3D`

u=y^2-x

           2
     -x + y

grd=Grad[u,Cartesian]


     {-1, 2 y, 0}

g1=PlotGradientField[u,{x,-3,3},{y,-3,3},PlotPoints->10];

g=Plot3D[u,{x,-3,3},{y,-3,3},Axes->Automatic,
PlotPoints->25,Shading->False]

c=ContourPlot[u,{x,-3,3},{y,-3,3},ContourShading->False,
Contours->10,PlotPoints->30];

Show[g1,c];

s=grd.grd

            2
     1 + 4 y

norm=grd/Sqrt[s]

              1               2 y
     {-(--------------), --------------, 0}
                    2                2
        Sqrt[1 + 4 y ]   Sqrt[1 + 4 y ]

Clear[x,y,z,u]

u2=17x-2x y/z+y^2 z^3

            2 x y    2  3
     17 x - ----- + y  z
              z

gradu2=Grad[u2,Cartesian]

           2 y  -2 x        3  2 x y      2  2
     {17 - ---, ---- + 2 y z , ----- + 3 y  z }
            z    z               2
                                z

g3=PlotGradientField3D[ u2,{x,-1,1},{y,-1,1},{z,1,2},
VectorHeads->True,PlotPoints->3,Axes->Automatic];

Show[g3,ViewPoint->{-4.000, -4.000, 4.000},Axes->Automatic]

PlotVectorField[{-x,y},{x,-1,1},{y,-1,1}];

PlotVectorField[{x,0},{x,-1,1},{y,-1,1},PlotPoints->7,
Axes->True];

PlotVectorField[{0,y},{x,-1,1},{y,-1,1},PlotPoints->7,
Axes->True];

PlotVectorField[{x,-y},{x,-1,1},{y,-1,1},PlotPoints->7,
Axes->True];

PlotVectorField[{-x,-y},{x,-1,1},{y,-1,1},PlotPoints->7,
Axes->True];

<<Graphics`PlotField3D`

PlotVectorField3D[{x,y,z},{x,-1,1},{y,-1,1},{z,-1,1},
VectorHeads->True,PlotPoints->4,Boxed->False,Axes->True];

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