(*^
::[	Information =

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:[font = section; inactive; Cclosed; noPageBreak; preserveAspect; cellOutline; backColorRed = 58981; backColorGreen = 58981; backColorBlue = 58981; startGroup]
Setup:
:[font = input; preserveAspect; endGroup]
Needs["Graphics`Graphics3D`"]
Needs["Graphics`Graphics`"]
Needs["Graphics`ParametricPlot3D`"]
:[font = section; inactive; Cclosed; noPageBreak; preserveAspect; cellOutline; backColorRed = 58981; backColorGreen = 58981; backColorBlue = 58981; startGroup]
Example 1: Representation of Complex Numbers:  (Simple warm-up)
:[font = input; noPageBreak; preserveAspect]
Clear["Global`*"];
:[font = text; inactive; preserveAspect]
Express the following quantities in the form "a + I b" where a and b are real numbers. 
(You might try and remember how you would do these by hand.)
;[s]
3:0,0;88,1;148,0;149,-1;
2:2,16,12,Times,0,14,0,0,0;1,16,12,Times,2,14,0,0,0;
:[font = text; inactive; preserveAspect]
1/(2 + 3 I) 
:[font = input; noPageBreak; preserveAspect]
1/(2 + 3 I) 
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1/(4 - 5 I) 
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1/(4 - 5 I) 
:[font = text; inactive; preserveAspect]
Sqrt[2 + 3 I] 
:[font = input; noPageBreak; preserveAspect]
Sqrt[2 + 3 I]  //N
:[font = text; inactive; preserveAspect]
Sqrt[2 + 3 I]: If you did it by hand, the answer would be: 
:[font = input; noPageBreak; preserveAspect]
(   Sqrt[Sqrt[13]] Cos[ArcTan[2,3]/2]
+ I Sqrt[Sqrt[13]] Sin[ArcTan[2,3]/2] ) 
:[font = text; inactive; preserveAspect]
Log[2 + 3 I] 
;[s]
2:0,0;13,1;14,-1;
2:1,16,12,Times,0,14,0,0,0;1,13,10,Courier,1,12,0,0,0;
:[font = input; noPageBreak; preserveAspect]
Log[2 + 3 I] //N
:[font = text; inactive; preserveAspect]
Log[2 + 3 I]: Again, doing it the long way we get: 
:[font = input; noPageBreak; preserveAspect]
Log[Sqrt[13]] + I ArcTan[2,3] //N
:[font = text; inactive; preserveAspect]
2 E^(-I Pi/3)
;[s]
2:0,0;13,1;14,-1;
2:1,16,12,Times,0,14,0,0,0;1,13,10,Courier,1,12,0,0,0;
:[font = input; noPageBreak; preserveAspect]
1/( 2 E^(I Pi/3) ) //N
:[font = text; inactive; preserveAspect]
2 E^(-I Pi/3): Again, doing it the long way we get: 
:[font = input; noPageBreak; preserveAspect]
(1/2) Cos[-Pi/3] + I (1/2) Sin[-Pi/3] //N
:[font = text; inactive; preserveAspect]
Sqrt[2 E^(I Pi/3) ]
;[s]
2:0,0;19,1;20,-1;
2:1,16,12,Times,0,14,0,0,0;1,13,10,Courier,1,12,0,0,0;
:[font = input; noPageBreak; preserveAspect]
Sqrt[2 E^(I Pi/3) ] //N
:[font = text; inactive; preserveAspect]
Sqrt[2 E^(I Pi/3) ]: Again, doing it the long way we get: 
:[font = input; noPageBreak; preserveAspect; endGroup]
Sqrt[2] Cos[Pi/3/2] + I Sqrt[2] Sin[Pi/3/2] //N
:[font = section; inactive; Cclosed; noPageBreak; preserveAspect; cellOutline; backColorRed = 58981; backColorGreen = 58981; backColorBlue = 58981; startGroup]
Problem 2: Branch cuts and discontinuties
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Clear["Global`*"];
:[font = subsection; inactive; Cclosed; noPageBreak; preserveAspect; startGroup]
Part a) Log[z]
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Let's consider the function f[z]= Log[z], where z=r Exp[I th].  We use CylindricalPlot3D to display the Re and Im part of this function in the complex plane.  Here is the real part:
:[font = input; noPageBreak; preserveAspect]
CylindricalPlot3D[
   Log[ r E^(I th) ] //Re
  ,{r,0,2}
  ,{th,-Pi ,+Pi}
  ,BoxRatios->{1,1,1}
  ];
:[font = text; inactive; preserveAspect]
Here is the Im part. Note, we have a branch cut starting at the origin, and running along the negative real axis. 
:[font = input; noPageBreak; preserveAspect]
CylindricalPlot3D[
   Log[ r E^(I th) ] //Im
  ,{r,0,2}
  ,{th,-Pi * 0.99,+Pi * 0.99}
  ,BoxRatios->{1,1,1}
  ];
:[font = text; inactive; preserveAspect]
Here, we compute the discontinuity across the branch cut. In this particular case, the discontinuity is independent of r. 
:[font = input; noPageBreak; preserveAspect]
((
  ( Log[ r E^( I Pi *a) ] 
  - Log[ r E^(-I Pi *a) ] 
  )  //Im  
     //PowerExpand 
 )   /.{a->1}
)
:[font = text; inactive; preserveAspect]
Here is a plot of the discontinuity vs. r. This is a profile of our 3-D curve above. 
:[font = input; noPageBreak; preserveAspect; endGroup]
Plot[
{Log[ r E^(I Pi  *0.999) ] 
,Log[ r E^(-I Pi *0.999) ] } //Im //Evaluate
,{r,0.001,5}];
:[font = text; inactive; preserveAspect; cellOutline]
Repeat for the following functions: 
:[font = subsection; inactive; noPageBreak; preserveAspect]
Part b) Sqrt[z]
:[font = subsection; inactive; noPageBreak; preserveAspect]
Part c) ArcSin[z]
:[font = subsection; inactive; noPageBreak; preserveAspect]
Part d) Sqrt[1-z^2]
:[font = subsection; inactive; noPageBreak; preserveAspect; endGroup]
Part e) ArcCos[z]
:[font = section; inactive; Cclosed; noPageBreak; preserveAspect; cellOutline; backColorRed = 58981; backColorGreen = 58981; backColorBlue = 58981; startGroup]
Problem 3: Visualization of different coordinate systems
:[font = text; inactive; preserveAspect]
Mathematica has a large number of built-in coordinate systems. A common problem is how to help the students visualize them.  We present one method below. 
;[s]
2:0,1;11,0;155,-1;
2:1,16,12,Times,0,14,0,0,0;1,16,12,Times,2,14,0,0,0;
:[font = section; inactive; Cclosed; preserveAspect; cellOutline; startGroup]
Load Package:
:[font = input; preserveAspect]
Clear["Global`*"]
:[font = text; inactive; preserveAspect]
We'll use the package: Calculus`VectorAnalysis`
;[s]
2:0,0;23,1;48,-1;
2:1,16,12,Times,0,14,0,0,0;1,13,10,Courier,1,12,0,0,0;
:[font = input; preserveAspect]
Needs["Calculus`VectorAnalysis`"]
:[font = text; inactive; preserveAspect]
Let's take a look at the built-in functions in  Calculus`VectorAnalysis`
;[s]
2:0,0;48,1;73,-1;
2:1,16,12,Times,0,14,0,0,0;1,13,10,Courier,1,12,0,0,0;
:[font = input; Cclosed; preserveAspect; startGroup]
?Calculus`VectorAnalysis`*
:[font = info; inactive; preserveAspect; endGroup; endGroup]
ArcLengthFactor          Div                      phi
Biharmonic               DotProduct               ProlateSpheroidal
Bipolar                  EllipticCylindrical      r
Bispherical              eta                      ScalarTripleProduct
Cartesian                Grad                     ScaleFactors
ConfocalEllipsoidal      JacobianDeterminant      SetCoordinates
ConfocalParaboloidal     JacobianMatrix           Spherical
Conical                  lambda                   theta
CoordinateRanges         Laplacian                Toroidal
Coordinates              mu                       u
CoordinatesFromCartesian nu                       v
CoordinatesToCartesian   OblateSpheroidal         x
CoordinateSystem         ParabolicCylindrical     xi
CrossProduct             Paraboloidal             y
Curl                     ParameterRanges          z
Cylindrical              Parameters
:[font = text; inactive; preserveAspect]
To get started correctly, we'll begin with a coordinate system where we know the correct answer. 
:[font = section; inactive; Cclosed; preserveAspect; cellOutline; startGroup]
Cartesian:
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SetCoordinates[Cartesian]
:[font = input; preserveAspect]
Coordinates[Cartesian]
:[font = input; preserveAspect]
CoordinateRanges[Cartesian]
:[font = text; inactive; preserveAspect]
We begin with a ContourPlot of the x coordinate in the {x,y} plane.  We are restricted to taking 2-D slices, so we will have to use three separate 2-D slices to get the full picture. 
:[font = input; preserveAspect]
p1= ContourPlot[x,{x,-3,3},{y,-3,3}
                 ,ContourShading->False
                 ,ContourStyle->{{RGBColor[1,0,0]}}
                 ];
:[font = text; inactive; preserveAspect]
Same as above, but this shows the y-variable in the {x,y} plane. 
(Note, we skip the z-variable in the {x,y} plane as this is uninteresting--try it.)
;[s]
3:0,0;66,1;149,0;150,-1;
2:2,16,12,Times,0,14,0,0,0;1,16,12,Times,2,14,0,0,0;
:[font = input; preserveAspect]
p2= ContourPlot[y,{x,-3,3},{y,-3,3}
                 ,ContourShading->False
                 ,ContourStyle->{{RGBColor[0,1,0]}}
                 ];
:[font = text; inactive; preserveAspect]
Putting these together, we obtain (not suprisingly), a Cartesian grid. 
:[font = input; preserveAspect]
Show[{p1,p2}];
:[font = text; inactive; preserveAspect]
We can now continue working in different 2-D planes. 
:[font = input; preserveAspect]
p1= ContourPlot[y,{y,-3,3},{z,-3,3}
                 ,ContourShading->False
                 ,ContourStyle->{{RGBColor[0,1,0]}}
                 ,DisplayFunction->Identity
                 ];
:[font = input; preserveAspect]
p2= ContourPlot[z,{y,-3,3},{z,-3,3}
                 ,ContourShading->False
                 ,ContourStyle->{{RGBColor[0,0,1]}}
                 ,DisplayFunction->Identity
                 ];
:[font = text; inactive; preserveAspect]
and the output (not shown) is again a Cartesian grid. 
:[font = input; preserveAspect; endGroup]
Show[{p1,p2},DisplayFunction->$DisplayFunction];
:[font = text; inactive; preserveAspect]
Let's try something not so trivial:
:[font = input; preserveAspect]
coordSys=Spherical;
:[font = section; inactive; Cclosed; preserveAspect; cellOutline; startGroup]
Spherical:
:[font = input; preserveAspect]
SetCoordinates[coordSys]
:[font = input; preserveAspect]
CoordinateRanges[]
:[font = input; preserveAspect]
coords= CoordinatesFromCartesian[{x,y,z}]
:[font = input; preserveAspect]
color={RGBColor[1,0,0],RGBColor[0,1,0],RGBColor[0,0,1]};
:[font = input; preserveAspect]
Do[

p[i]= ContourPlot[coords[[i]] /.{z->0} //Evaluate
                 ,{x,-3,3},{y,-3,3}
                 ,ContourShading->False
                 ,ContourStyle->{{ color[[i]] }}
                 ,DisplayFunction->Identity
                 ];
                 
,{i,1,3}];

Show[{p[1],p[2],p[3]},DisplayFunction->$DisplayFunction];
:[font = input; preserveAspect]
Do[

p[i]= ContourPlot[coords[[i]] /.{x->0} //Evaluate
                 ,{y,-3,3},{z,-3,3}
                 ,ContourShading->False
                 ,ContourStyle->{{ color[[i]] }}
                 ,DisplayFunction->Identity
                 ];
                 
,{i,1,3}];

Show[{p[1],p[2],p[3]},DisplayFunction->$DisplayFunction];
:[font = input; preserveAspect; endGroup]
Do[

p[i]= ContourPlot[coords[[i]] /.{y->0} //Evaluate
                 ,{x,-3,3},{z,-3,3}
                 ,ContourShading->False
                 ,ContourStyle->{{ color[[i]] }}
                 ,DisplayFunction->Identity
                 ];
                 
,{i,1,3}];

Show[{p[1],p[2],p[3]},DisplayFunction->$DisplayFunction];
:[font = input; preserveAspect]
coordSys=Bispherical;
:[font = section; inactive; Cclosed; preserveAspect; cellOutline; startGroup]
Bispherical:
:[font = input; preserveAspect]
SetCoordinates[coordSys]
:[font = input; preserveAspect]
CoordinateRanges[]
:[font = input; preserveAspect]
coords= CoordinatesFromCartesian[{x,y,z}]
:[font = input; preserveAspect]
color={RGBColor[1,0,0],RGBColor[0,1,0],RGBColor[0,0,1]};
:[font = input; preserveAspect]
Do[

p[i]= ContourPlot[coords[[i]] /.{z->0} //Evaluate
                 ,{x,-3,3},{y,-3,3}
                 ,ContourShading->False
                 ,ContourStyle->{{ color[[i]] }}
                 ,DisplayFunction->Identity
                 ];
                 
,{i,1,3}];

Show[{p[1],p[2],p[3]},DisplayFunction->$DisplayFunction];
:[font = input; preserveAspect]
Do[

p[i]= ContourPlot[coords[[i]] /.{x->0} //Evaluate
                 ,{y,-3,3},{z,-3,3}
                 ,ContourShading->False
                 ,ContourStyle->{{ color[[i]] }}
                 ,DisplayFunction->Identity
                 ];
                 
,{i,1,3}];

Show[{p[1],p[2],p[3]},DisplayFunction->$DisplayFunction];
:[font = input; preserveAspect; endGroup]
Do[

p[i]= ContourPlot[coords[[i]] /.{y->0} //Evaluate
                 ,{x,-3,3},{z,-3,3}
                 ,ContourShading->False
                 ,ContourStyle->{{ color[[i]] }}
                 ,DisplayFunction->Identity
                 ];
                 
,{i,1,3}];

Show[{p[1],p[2],p[3]},DisplayFunction->$DisplayFunction];
:[font = input; preserveAspect]
coordSys=ConfocalParaboloidal;
:[font = section; inactive; preserveAspect; cellOutline]
ConfocalParaboloidal:
:[font = text; inactive; preserveAspect; fontColorRed = 65535]
Copy from above and repeat.
:[font = input; preserveAspect]
coordSys=ConfocalEllipsoidal;
:[font = section; inactive; preserveAspect; cellOutline]
ConfocalEllipsoidal:
:[font = text; inactive; preserveAspect; fontColorRed = 65535]
Copy from above and repeat.
:[font = section; inactive; preserveAspect; cellOutline]
You try your own here: ....
:[font = text; inactive; preserveAspect; fontColorRed = 65535; endGroup]
Copy from above and repeat.
^*)