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A brief discussions on derivative and the"Gradient"

H.Tahsiri

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,
```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},

`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},