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Physics 4321
Homework 2
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P
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closepath
clip
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% End of Graphics
MathPictureEnd

:[font = section; inactive; preserveAspect; startGroup]
Example 1: Cylindrical Coordinates:
You do it in Spherical
;[s]
2:0,0;36,1;59,-1;
2:1,19,14,Times,1,18,0,0,0;1,16,12,Times,1,14,0,0,0;
:[font = input; preserveAspect]
Clear["Global`*"]
:[font = input; preserveAspect; startGroup]
Coordinates[Cartesian]
:[font = output; output; inactive; preserveAspect; endGroup]
{x, y, z}
;[o]
{x, y, z}
:[font = input; preserveAspect; startGroup]
Coordinates[Cylindrical]
:[font = output; output; inactive; preserveAspect; endGroup]
{r, theta, z}
;[o]
{r, theta, z}
:[font = input; preserveAspect; startGroup]
CoordinatesFromCartesian[{x,y,z},Cylindrical]
:[font = output; output; inactive; preserveAspect; endGroup]
{(x^2 + y^2)^(1/2), ArcTan[x, y], z}
;[o]
       2    2
{Sqrt[x  + y ], ArcTan[x, y], z}
:[font = input; preserveAspect; startGroup]
CoordinatesToCartesian[{r, theta, z},Cylindrical]
:[font = output; output; inactive; preserveAspect; endGroup]
{r*Cos[theta], r*Sin[theta], z}
;[o]
{r Cos[theta], r Sin[theta], z}
:[font = input; preserveAspect; startGroup]
x2rRule = Thread[ {x,y,z} -> CoordinatesToCartesian[{r, theta, z},Cylindrical] ]
x2rRule = x2rRule //Drop[#,-1]&
:[font = output; output; inactive; preserveAspect; startGroup]
{x -> r*Cos[theta], y -> r*Sin[theta], z -> z}
;[o]
{x -> r Cos[theta], y -> r Sin[theta], z -> z}
:[font = output; output; inactive; preserveAspect; endGroup; endGroup]
{x -> r*Cos[theta], y -> r*Sin[theta]}
;[o]
{x -> r Cos[theta], y -> r Sin[theta]}
:[font = input; preserveAspect; startGroup]
r2xRule = Thread[ {r, theta, z} -> CoordinatesFromCartesian[{x,y,z},Cylindrical] ]
r2xRule = r2xRule //Drop[#,-1]&
:[font = output; output; inactive; preserveAspect; startGroup]
{r -> (x^2 + y^2)^(1/2), theta -> ArcTan[x, y], z -> z}
;[o]
            2    2
{r -> Sqrt[x  + y ], theta -> ArcTan[x, y], z -> z}
:[font = output; output; inactive; preserveAspect; endGroup; endGroup]
{r -> (x^2 + y^2)^(1/2), theta -> ArcTan[x, y]}
;[o]
            2    2
{r -> Sqrt[x  + y ], theta -> ArcTan[x, y]}
:[font = input; preserveAspect; startGroup]
JacobianDeterminant[{x,y,z},Cartesian]
:[font = output; output; inactive; preserveAspect; endGroup]
1
;[o]
1
:[font = input; preserveAspect; startGroup]
JacobianDeterminant[{r,theta,z},Cylindrical]
:[font = output; output; inactive; preserveAspect; endGroup]
r
;[o]
r
:[font = input; preserveAspect]
Integrate[ 1 JacobianDeterminant[{r,theta,z},Cylindrical]
   ,{theta,0, 2 Pi}
   ,{z,0,h}
   ]
:[font = input; preserveAspect; startGroup]
Integrate[ 1 JacobianDeterminant[{r,theta,z},Cylindrical]
   ,{r,0,R}
   ,{theta,0, 2 Pi}
   ,{z,0,h}
   ]
:[font = output; output; inactive; preserveAspect; endGroup; endGroup]
h*Pi*R^2
;[o]
      2
h Pi R
:[font = section; inactive; preserveAspect; startGroup]
Example 2:
:[font = input; preserveAspect]
Clear["Global`*"]
:[font = input; preserveAspect]
v={y,-x,0};
:[font = input; preserveAspect; startGroup]
x2rRule = Thread[ {x,y,z} -> CoordinatesToCartesian[{r, theta, z},Cylindrical] ]
x2rRule = x2rRule //Drop[#,-1]&
:[font = output; output; inactive; preserveAspect; startGroup]
{x -> r*Cos[theta], y -> r*Sin[theta], z -> z}
;[o]
{x -> r Cos[theta], y -> r Sin[theta], z -> z}
:[font = output; output; inactive; preserveAspect; endGroup; endGroup]
{x -> r*Cos[theta], y -> r*Sin[theta]}
;[o]
{x -> r Cos[theta], y -> r Sin[theta]}
:[font = input; preserveAspect]
  rUnit={ x,y,0}/Sqrt[x^2+y^2];
phiUnit={-y,x,0}/Sqrt[x^2+y^2];
  zUnit={ 0,0,1};
:[font = input; preserveAspect; startGroup]
vr= {rUnit,phiUnit,zUnit}.v  /.x2rRule //Simplify //PowerExpand
:[font = output; output; inactive; preserveAspect; endGroup]
{0, -r, 0}
;[o]
{0, -r, 0}
:[font = input; preserveAspect; startGroup]
PlotVectorField[ v[[{1,2}]] ,{x,-1,1},{y,-1,1},PlotPoints->4];
:[font = postscript; PostScript; formatAsPostScript; output; inactive; preserveAspect; pictureLeft = 34; pictureWidth = 182; pictureHeight = 182; endGroup]
%!
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grestore
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