(*^
::[	Information =

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Setup:
:[font = input; preserveAspect; endGroup]
Clear["Global`*"];
Off[ General::spell ];
Off[ General::spell1];
Off[Clear::wrsym];
:[font = section; inactive; Cclosed; noPageBreakInGroup; preserveAspect; cellOutline; backColorRed = 58981; backColorGreen = 58981; backColorBlue = 58981; startGroup]
Load Packages:
:[font = input; preserveAspect; endGroup]
Needs["LinearAlgebra`Orthogonalization`"]
Needs["Calculus`VectorAnalysis`"]
Needs["Graphics`PlotField`"]
Needs["Graphics`PlotField3D`"]

:[font = section; inactive; Cclosed; noPageBreakInGroup; preserveAspect; cellOutline; backColorRed = 58981; backColorGreen = 58981; backColorBlue = 58981; startGroup]
Initialization: 
Define Home-made: VectorPlot, VectorPlot3D VectorLength
:[font = input; preserveAspect]
Unprotect[{VectorPlot,VectorPlot3D,VectorLength}];
:[font = input; preserveAspect]
VectorPlot[list__List?VectorQ]:= VectorPlot[{list}];

VectorPlot[list_List?MatrixQ]:= 
Module[{length,totList},
 length=Length[list];
 totList=Table[{{0,0},list[[i]]} ,{i,1,length}];
 Return[
  Show[Graphics[{Thickness[0.015],Line /@ totList }
          ,{GridLines->Automatic
           ,Axes->True
           ,AspectRatio->1}]]
 ]
];
:[font = input; preserveAspect]
VectorPlot3D[list__List?VectorQ]:= VectorPlot3D[{list}];

VectorPlot3D[list_List?MatrixQ]:= 
Module[{length,totList},
 length=Length[list];
 totList=Table[{{0,0,0},list[[i]]} ,{i,1,length}];
 Return[
  Show[Graphics3D[{Thickness[0.015],Line /@ totList}
          ,{ FaceGrids->{{0,0,-1},{0,1,0},{-1,0,0}}
            ,BoxRatios->{1,1,1}
            ,Axes->True}]]
 ]
];
:[font = input; preserveAspect]

VectorLength[vector_List?VectorQ]:=Sqrt[vector.vector]
:[font = input; preserveAspect; endGroup]
Protect[{VectorPlot,VectorPlot3D,VectorLength}];
:[font = section; inactive; Cclosed; noPageBreakInGroup; preserveAspect; cellOutline; backColorRed = 58981; backColorGreen = 58981; backColorBlue = 58981; startGroup]
Example 1: Plotting and adding vectors
:[font = text; inactive; preserveAspect]
This is a simple exercise to help the student see the relationship between the graphical and analytic representation of vectors.  The exercise is to plot {a,b,c}, and estimate the result BEFORE plotting {d}. 
:[font = input; preserveAspect]
Clear["Global`*"]
:[font = input; preserveAspect]
a= 3 {Cos[ 30 Degree],Sin[ 30 Degree]} //N
b= 5 {Cos[ 50 Degree],Sin[ 50 Degree]} //N
c= 7 {Cos[170 Degree],Sin[170 Degree]} //N
VectorLength /@ {a,b,c}  
:[font = input; preserveAspect]
d=a+b+c
:[font = input; preserveAspect; endGroup]
VectorPlot[a,b,c,d];
:[font = section; inactive; Cclosed; noPageBreakInGroup; preserveAspect; cellOutline; backColorRed = 58981; backColorGreen = 58981; backColorBlue = 58981; startGroup]
Example 2: Dot and Cross products of vectors
:[font = text; inactive; preserveAspect]
This is a simple exercise in computing vector products, both scalar and vector. 
:[font = input; preserveAspect]
Clear["Global`*"]
:[font = text; inactive; preserveAspect]
Given the following vectors, we compute requested quantities. 
:[font = input; preserveAspect]
a={3,2,-1};
b={1,2, 3};
c={1,0, 1};
:[font = subsection; inactive; noPageBreakInGroup; preserveAspect]
Part a)
:[font = input; preserveAspect]
{a.b, a.c, b.c}
:[font = subsection; inactive; noPageBreakInGroup; preserveAspect]
Part b)
:[font = input; preserveAspect]
{CrossProduct[a,b]
,CrossProduct[a,c]
,CrossProduct[b,c]
}
:[font = input; preserveAspect]
{CrossProduct[b,a]
,CrossProduct[c,a]
,CrossProduct[c,b]
}
:[font = subsection; inactive; noPageBreakInGroup; preserveAspect]
Part c)
:[font = input; preserveAspect]
ArcCos[a.b /(VectorLength[a] VectorLength[b]) ]/Degree //N
:[font = input; preserveAspect]
ArcCos[b.c /(VectorLength[b] VectorLength[c]) ]/Degree //N
:[font = input; preserveAspect]
ArcCos[a.c /(VectorLength[a] VectorLength[c]) ]/Degree //N
:[font = subsection; inactive; noPageBreakInGroup; preserveAspect]
Part d)
:[font = input; preserveAspect]
(a+b).c
:[font = subsection; inactive; noPageBreakInGroup; preserveAspect]
Part e)
:[font = input; preserveAspect]
{a.CrossProduct[b,c] 
,b.CrossProduct[c,a] 
,c.CrossProduct[a,b] 
}
:[font = subsection; inactive; noPageBreakInGroup; preserveAspect]
Part f)
:[font = input; preserveAspect; endGroup]
{CrossProduct[a,CrossProduct[b,c] ]
,CrossProduct[b,CrossProduct[c,a] ]
,CrossProduct[c,CrossProduct[a,b] ]
}
:[font = section; inactive; Cclosed; noPageBreakInGroup; preserveAspect; cellOutline; backColorRed = 58981; backColorGreen = 58981; backColorBlue = 58981; startGroup]
Example 3: Gradient of a scalar field
:[font = text; inactive; preserveAspect]
This exercise shows the students the relationship between  a scalar function and the gradient.
:[font = input; preserveAspect]
Clear["Global`*"]
:[font = input; preserveAspect]
phi= -( x^2+y^2 );
:[font = input; preserveAspect]
Plot3D[ phi  
      ,{x,-1,1}
      ,{y,-1,1}
      ,BoxRatios->{1,1,1}
      ];
:[font = input; preserveAspect]
ContourPlot[ phi  
      ,{x,-1,1}
      ,{y,-1,1}
      ];
:[font = input; preserveAspect]
PlotGradientField[ phi
   ,{x,-1,1}
   ,{y,-1,1}
   ,VectorHeads->True
   ,PlotPoints->8
   ,Axes->True
   ];
:[font = text; inactive; preserveAspect]
Having seen the picture of the gradient field, we compute the analytic result, and compare with the figure. 
:[font = input; preserveAspect]
Grad[phi]
:[font = input; preserveAspect]
Grad[phi][[{1,2}]]
:[font = input; preserveAspect]
PlotVectorField[ Grad[phi][[{1,2}]] //Evaluate
   ,{x,-1,1}
   ,{y,-1,1}
   ,VectorHeads->True
   ,PlotPoints->8
   ,Axes->True
   ];
:[font = text; inactive; preserveAspect]
Here is a 3-D version.
:[font = input; preserveAspect]
phi= -( x^2+y^2+z^2 );
:[font = input; preserveAspect; endGroup]
PlotGradientField3D[ phi
   ,{x,0,1}
   ,{y,0,1}
   ,{z,0,1}
   ,VectorHeads->True
   ,PlotPoints->4
   ,Axes->True
   ];
:[font = section; inactive; Cclosed; noPageBreakInGroup; preserveAspect; cellOutline; backColorRed = 58981; backColorGreen = 58981; backColorBlue = 58981; startGroup]
Example 4: Divergence of a vector field  I
:[font = text; inactive; preserveAspect]
This is a function that has a non-zero divergence. Again, this allows the students to gain an intuitive feeling for the vector calculus operations. 
:[font = input; preserveAspect]
Clear["Global`*"]
:[font = input; preserveAspect]
v={x,0};
:[font = input; preserveAspect]
Div[{x,0,0}]  (* needs to be 3-D for Mathematica *)
;[s]
3:0,0;37,1;48,0;52,-1;
2:2,12,10,Courier,1,12,0,0,0;1,12,10,Courier,3,12,0,0,0;
:[font = input; preserveAspect; endGroup]
PlotVectorField[v
     ,{x,-1,1}
     ,{y,-1,1}
     ,PlotPoints->8
     ];
:[font = section; inactive; Cclosed; noPageBreakInGroup; preserveAspect; cellOutline; backColorRed = 58981; backColorGreen = 58981; backColorBlue = 58981; startGroup]
Example 5: Divergence of a vector field  II
:[font = text; inactive; preserveAspect]
In contrast to the above, this is a function that has a zero divergence. 
Again, this allows the students to gain an intuitive feeling for the vector calculus operations. 
:[font = input; preserveAspect]
Clear["Global`*"]
:[font = input; preserveAspect]
v={1,2};
:[font = input; preserveAspect]
Div[{1,2,0}]  (* needs to be 3-D for Mathematica *)
;[s]
3:0,0;37,1;48,0;52,-1;
2:2,12,10,Courier,1,12,0,0,0;1,12,10,Courier,3,12,0,0,0;
:[font = input; preserveAspect; endGroup]
PlotVectorField[v
     ,{x,-1,1}
     ,{y,-1,1}
     ,PlotPoints->8
     ];
:[font = section; inactive; Cclosed; noPageBreakInGroup; preserveAspect; cellOutline; backColorRed = 58981; backColorGreen = 58981; backColorBlue = 58981; startGroup]
Example 6: {Div,Curl}={0,-2}
:[font = text; inactive; preserveAspect]
Here is a function that has a non-zero curl and zero divergence.
:[font = input; preserveAspect]
Clear["Global`*"]
:[font = input; preserveAspect]
v={y,-x,0};
:[font = input; preserveAspect]
Div[v]  
:[font = input; preserveAspect]
Curl[v]  
:[font = input; preserveAspect]
PlotVectorField[v[[{1,2}]]
     ,{x,-1,1}
     ,{y,-1,1}
     ,PlotPoints->8
     ];
:[font = text; inactive; preserveAspect]
Here is a 3-D rendering.
:[font = input; preserveAspect; endGroup]
PlotVectorField3D[v 
     ,{x,-1,1}
     ,{y,-1,1}
     ,{z,-1,1}
     ,PlotPoints->3
     ,VectorHeads->True
     ,ViewPoint->{1,2,4}
     ];
:[font = section; inactive; Cclosed; noPageBreakInGroup; preserveAspect; cellOutline; backColorRed = 58981; backColorGreen = 58981; backColorBlue = 58981; startGroup]
Example 7: {Div,Curl}={2,0}
:[font = text; inactive; preserveAspect]
Here is a function that has a zero curl and non-zero divergence.
:[font = input; preserveAspect]
Clear["Global`*"]
:[font = input; preserveAspect]
v={x,y,0};
:[font = input; preserveAspect]
Div[v]
:[font = input; preserveAspect]
Curl[v] 
:[font = input; preserveAspect]
PlotVectorField[v[[{1,2}]]
     ,{x,-1,1}
     ,{y,-1,1}
     ,PlotPoints->8
     ];
:[font = text; inactive; preserveAspect]
Here is a 3-D rendering.
:[font = input; preserveAspect; endGroup]
PlotVectorField3D[v 
     ,{x,-1,1}
     ,{y,-1,1}
     ,{z,-1,1}
     ,PlotPoints->3
     ,VectorHeads->True
     ,ViewPoint->{1,2,4}
     ];
:[font = section; inactive; noPageBreakInGroup; preserveAspect; cellOutline; backColorRed = 58981; backColorGreen = 58981; backColorBlue = 58981; startGroup]
Problem 8: {Div,Curl}={2,1}
:[font = text; inactive; Cclosed; preserveAspect; fontColorRed = 65535; startGroup]
Can you find  a function that has a zero curl and non-zero divergence.
:[font = input; preserveAspect]
Clear["Global`*"]
:[font = input; preserveAspect]
v={0,0,0};
:[font = input; preserveAspect]
Div[v]
:[font = input; preserveAspect]
Curl[v] 
:[font = input; preserveAspect]
PlotVectorField[v[[{1,2}]]
     ,{x,-1,1}
     ,{y,-1,1}
     ,PlotPoints->8
     ];
:[font = text; inactive; preserveAspect]
Here is a 3-D rendering.
:[font = input; preserveAspect; endGroup; endGroup]
PlotVectorField3D[v 
     ,{x,-1,1}
     ,{y,-1,1}
     ,{z,-1,1}
     ,PlotPoints->3
     ,VectorHeads->True
     ,ViewPoint->{1,2,4}
     ];
^*)