GraphingCalculator 4;
Window 45 2 799 1156;
PaneDivider 407;
FontSizes 14;
BackgroundType 0;
BackgroundColor 255 255 255;
StackPanes 1;
UseAntialiasing 0;
U -1 1;
V -1 1;
3D.X -4 4;
3D.Y -4 4;
3D.Z -4 4;
3D.Axes 0;
3D.Depth 0.5930909742999999;
3D.View 0.3399415787415959 0.9399086702111976 -0.03180274790679882 -0.9203329212058666 0.3394373419077572 0.194344038069522 0.1934606865935121 -0.03679650323564786 0.9804177579798893;
3D.Speed 0;
Text "<size=16>Fux through an orientable area and clsoed surface.   Corresponding flux tube.

Plots a vector field W in the x-direction.  Plots a square ""frame"" with unit normal N at spherical angles (theta,phi)=(a,b) and passing through a point X0 a distance d from the origin.  Also plots a closed surface.

Version 0.31  11-9-15
To do:
(i) Add area element on closed surface
(ii) ...


Unit normal:";
Color 2;
Expr N=vector(sin(a)*cos(b),sin(a)*sin(b),cos(a));
Text "<size=16>Frame passes through this point:";
Color 3;
Opacity 0.7;
Expr X_0=d*N;
Text "<size=16>A point along the normal direction a distance f from the origin:";
Expr f=slider([1,10,40]);
Color 7;
Expr R=f*N;
Text "<size=16>T and P are the spherical theta-hat and phi-hat unit vectors:";
Color 4;
Expr T=vector(cos(a)*cos(b),cos(a)*sin(b),-sin(a)),P=vector(-sin(b),cos(b),0);
Color 5;
Expr O=vector(0,0,0),I=vector(1,0,0),J=vector(0,1,0),K=vector(0,0,1);
Text "<size=16>U and V are unit vectors normal to N that define the plane of the frame, rotated by an angle g about N if we wish to adjust the orientation of the frame.";
Color 3;
Expr U=T*cos(g)+P*sin(g);
Color 5;
Expr V=-(T*sin(g))+P*cos(g);
Text "<size=16>A square frame X0+uU+vV passing through X0 and oriented at an angle g about N:";
Color 17;
Expr vector(x,y,z)=X_0+U*u+V*v;
Text "<size=16>A vector field along the x-direction representing a flow of some kind:";
Color 7;
MathPaneSlider 5;
Expr W=slider([0,10,40]);
Color 3;
Grain 0.2;
Expr vector(d*x,d*y,d*z)=W*vector(1,0,0);
Color 8;
Expr h=slider([0,0.05]);
Color 17;
Expr X_0,X_0+R,'radius'=h;
Text "<size=16>A ""flux tube"" illustrating the portion of the vector flow that passes through the frame.  In its current form, only works properly when  a and b are varied separately and d=0.  [Tube is 0x+(y/cosb)^m+(z/sina)^m=1; shift center and orientation to generalize; slows down plotting significantly.]";
Grain 1;
Opacity 0.7;
Expr 0*[x]^m+[y/cos(b)]^m+[z/sin(a)]^m=1;
Opacity 0.7;
Expr m=10;
Text "<size=16>(a,b) are (theta,phi) of unit normal to frame N; d is the distance from the origin to the place; g is the angle of rotation of the frame about N.";
Color 4;
Expr G=slider([0,1,40]);
Color 2;
Expr d=slider([0,10]);
Color 8;
Expr a=pi*A,b=2*pi*B,g=2*pi*G;
Color 6;
MathPaneSlider 20;
Expr A=slider([0,1,40]);
Expr B=slider([0,1,40]);
Text "<size=16>Flux through a closed surface.";
Color 6;
Expr M=slider([1,10,9]);
Color 17;
Grain 0.5166666666666667;
Expr x^(2*M)+y^(2*M)+z^(2*M)=1;
Text "<size=16>Normals to cube (M>1):";
Color 17;
Expr O,2*I,'radius'=h;
Color 17;
Expr O,-(2*I),'radius'=h;
Color 17;
Expr O,2*J,'radius'=h;
Color 17;
Expr O,-(2*J),'radius'=h;
Color 17;
Expr O,2*K,'radius'=h;
Color 17;
Expr O,-(2*K),'radius'=h;
Text "<size=16>Normal to sphere (M=1):";
Color 17;
Expr O,2*N,'radius'=h;
Text "<size=20>
<size=13>Author: David A. Craig <<http://web.lemoyne.edu/~craigda/>";