ÁëëáãÞ ìåôáâëçôþí : ÊõëéíäñéêÝò êáé ÓöáéñéêÝò óõíôåôáãìÝíåò

NÂ Init: ( Ìçí áããßæåôå)

Mathematica 2. Initializations
Book 3

Needs["Graphics`Colors`"]
Off[General::spell];
Off[General::spell1];
Off[ParametricPlot3D::ppcom];
Off[ParametricPlot::ppcom];
Off[Plot::plnr];
SetOptions[Limit,Analytic->True];
Off[Clear::ssym];

CMView = {2.7, 1.6, 1.2};

ThreeAxes[u,v]

ThreeAxes[u_,v_] := 
      Graphics3D[{
            {Blue,Line[{{-u,0,0},{u,0,0}}]},
                    Text["x",{u + v,0,0}],
            {Blue,Line[{{0,-u,0},{0,u,0}}]},
                    Text["y",{0, u + v,0}],
            {Blue,Line[{{0,0,0},{0,0,u}}]},
                    Text["z",{0,0,u + v}]}]; 
ThreeAxes[u_] := ThreeAxes[u,u/8];
ThreeAxes::usage = "ThreeAxes[a,b] makes a standard cartesian axis graphics object
	with x, y, and z running from -a to a, and with axis
	labels b units beyond the tips of the axes.  ThreeAxes[a]
	is ThreeAxes[a,a/8].";

Vector Drawers

BeginPackage["Vectors3D`","Graphics`Colors`"];

(* Last repair, February, 1993,  Bill Davis   *)

Perp::usage = "Perp[expr] returns a single 3-d vector which is 
		perpendicular to the vector, 'expr'.";
		
Cross::usage = "Cross[a,b] returns the cross product of a and b.";

Norm::usage = "Norm[expr] returns the Euclidean length of the
		vector, 'expr'.";
		
Turn::usage = "Turn[expr] returns an orthogonal 3 by 3 matrix 
		which takes {0,0,1} to 'expr'.";
		
Vector::notnum = "The vectors `1` and `2` aren't numeric.";
		
Arrow::usage = "Arrow[expr] produces a 2 or 3 dimensional 
	vector from the origin to expr. Arrow[expr,Tail->tail] gives a 
	vector from tail to expr.  RGBColor and Thickness may be added as 
	options";
	
Vector::usage = "Vector[a,b] gives a vector from a to b. 
			RGBColor and Thickness may be added as options";

Vector2D::usage = "Vector2D[a,b] produces a 2-d graphics object, a 
  		vector from a to b. Vector[a,b,options] produces the 
  		graphics object with the styles such as RGBColor and 
  		Thickness included.";
		
Vector3D::usage = "Vector3D[a,b] produces the 3-d graphics object,
		a vector from a to b.  Vector3D[a,b,options] produces the 
		graphics object with the options such as RGBColor and 
		Thickness included.";
		
Begin["`Private`"];

SetOptions[ParametricPlot,AspectRatio->Automatic];
SetOptions[Plot,AspectRatio->Automatic];
SetOptions[Graphics,AspectRatio->Automatic];

colorQ[a_]:=(Head[a]===RGBColor);
grayQ[a_]:=(Head[a]===GrayLevel);
thickQ[a_]:=(Head[a]===Thickness);
notruleQ[u_] := !(Head[u]===Rule);
ruleQ[u_] := (Head[u]===Rule);

Perp[a_?VectorQ] := If[a[[1]]==0,Return[{1,0,0}],
			 If[a[[2]]==0,Return[{0,1,0}],
			  Return[{-a[[2]],a[[1]],0}]]];
			
				
Norm[a_?VectorQ]:=Sqrt[a.a];

Turn[a_?VectorQ] := Module[{A = a, B},
		B = Perp[A]/Norm[Perp[A]];
		A = a/Norm[a];
		Return[Transpose[{B,A~Cross~B,A}]]
					];
					
corn = Table[N[1/4 {1.3Cos[2 k Pi/7],1.3Sin[2 k Pi/7],-4}],
	{k,0,9}]; 

corners[a_] := N[Table[(a.corn[[j]]),{j,1,10}]]; 

head[A_,a_,b_,hr___] := 
	Module[{HR},
	HR=ToExpression["HeadRatio"]/.Select[hr,ruleQ]/.
		(ToExpression["HeadRatio"]->.17);
	Return[Flatten[
		Table[{b,b+HR Norm[b-a] corners[A][[j-1]],
			b + HR Norm[b-a] corners[A][[j]]},{j,2,9}],1]]];
				

Vector3D[a_,b_,opts___] := 
	Module[{A,B,AA,BB,m,aa,bb,bob,boob,booob},
		aa=N[a];bb=N[b];
		If[!NumberQ[aa.aa+bb.bb],
			Message[Vector::notnum,a,b];Return[Null],					
			If[Chop[aa-bb]=={0.,0.,0.},
				B={bb,bb},
				A = Turn[bb-aa];
				B = head[A,aa,bb,{opts}]];
		bob=Join[Select[{opts},colorQ],Select[{opts},thickQ],
			Select[{opts},grayQ]];
		boob=Join[Select[{opts},colorQ],Select[{opts},grayQ]];
		booob=If[Length[boob]==0,RGBColor[0,0,0],
				boob[[1]]];			
		Return[Graphics3D[{Join[bob,{Line[{aa,bb}]}],
			{SurfaceColor[booob],Polygon[B]}}]]
					]];
									
									

Vector2D[from_,to_,opts___] := Module[{A,B,AA,BB,c,d,a},
			AA=N[to];
			BB=N[from];
			
			If[!NumberQ[AA.AA+BB.BB],
				Message[Vector::notnum,from,to];
				Return[Null],
			A=AA-BB;
			
			a = Join[Select[{opts},colorQ],Select[{opts},
				thickQ],Select[{opts},grayQ]];
			
			If[Chop[AA-BB]=={0.,0.},
				Return[Graphics[{Join[a,{Line[{BB,AA}]}]}]],
			
			B = RotateLeft[A] {-1,1};
			e = ToExpression["HeadRatio"]/.
					Select[{opts},ruleQ]/.
				{ToExpression["HeadRatio"]->.21};
			c = AA - e A + e/3 B;
			d = AA - e A - e/3 B;
						
			Return[
			Graphics[{Join[a,{Line[{BB,AA}]}],
					Join[a,{Polygon[{AA,c,d}]}]}]]]]];
						
					
Vector[from_,to_,opts___]:=
	If[Length[N[from]]==2,Return[Vector2D[from,to,opts]],
		Return[Vector3D[from,to,s]]];
		
Arrow[vec_,opts___]:= 
	Module[{a,b},
				If[Length[N[vec]]==2,
				a=ToExpression["Tail"]/.Select[{opts},ruleQ]/.
					{ToExpression["Tail"]->{0,0}};
				Return[Vector2D[a,a+vec,opts]],
				a=ToExpression["Tail"]/.Select[{opts},ruleQ]/.
					{ToExpression["Tail"]->{0,0,0}};
				Return[Vector3D[a,a+vec,opts]]]
				];
	
	
(*Protect[Perp,Cross,Turn,Vector3D,Vector2D,Vector,Arrow];*)

End[];


EndPackage[];
Null

Spherical Setup

Clear[polar]
polar[{x_,y_,z_}]:=
{Sqrt[x^2+y^2+z^2],ArcTan[y/x],ArcCos[z/Sqrt[x^2+y^2+z^2]]};
Clear[X]
length[X_]:=Sqrt[X.X];
rr=3;
R=5;
scale=0.8;
red=RGBColor[1,0,0];
blue=RGBColor[0,0,1];
color1=RGBColor[0.8,0.2,0.8];
color2=RGBColor[0.2,0.8,0.2];
black=RGBColor[0,0,0];
Clear[axes]
axes:=Graphics3D[{
Line[{X,{1.5rr,0,0}}],
Line[{X,{0,1.5rr,0}}],
Line[{X,{0,0,2rr}}]}
];
Clear[xyplane]
xyplane:=Graphics3D[Polygon[
{
{-2rr,-2rr,0},
{-2rr, 2rr,0},
{ 2rr, 2rr,0},
{ 2rr,-2rr,0}
}]];
X={0,0,0};
origin=Graphics3D[{black,Point[X]}];
Z={2,3,5};
Clear[point]
point:=Graphics3D[{red,PointSize[0.03],Point[scale Z]}];
Z0={0,0,length[0.8 Z]};
Clear[zpoint]
zpoint=Graphics3D[{black,Point[Z0]}];
XY=scale Z {1,1,0};
Clear[xypoint]
xypoint:=Graphics3D[{black,Point[XY]}];
X0={length[scale Z {1,1,0}],0,0};
xpoint=Graphics3D[{black,Point[X0]}];
Y0={0,length[scale Z {1,1,0}],0};
ypoint=Graphics3D[{black,Point[Y0]}];
Clear[ray]
ray:=Graphics3D[{blue,Line[{X,scale Z}]}];
Clear[ray1]
ray1:=Graphics3D[{color1,Line[{scale Z,Z}]}];
Clear[xyray]
xyray:=Graphics3D[{red,Line[{X, Z {1,1,0}}]}];
rays:={ray,xyray};
xyaux=
length[0.8 Z](0.8 Z {1,1,0})/length[0.8 Z {1,1,0}];
Clear[zarc]
zarc=Graphics3D[{blue,Line[
Table[Cos[t]{0,0,length[0.8 Z]} + Sin[t]xyaux,
{t,0,polar[0.8 Z][[3]],.1}]
]}];
Clear[zarc1]
zarc1=Graphics3D[{color1,Line[
Table[Cos[t]{0,0,length[0.8 Z]} + Sin[t]xyaux,
{t,polar[0.8 Z][[3]],1,.1}]
]}];
Clear[zarc2]
zarc2=Graphics3D[{color1,Line[
Table[Cos[t]{0,0,length[0.8 Z]} + Sin[t]xyaux,
{t,-.8,0,.1}]
]}];
Clear[xyarc]
xyarc=Graphics3D[{blue,Line[
Table[Cos[t] X0 + Sin[t] Y0,
{t,0,polar[XY][[2]],.1}]
]}];
Clear[xyarc1]
xyarc1=Graphics3D[{color1,Line[
Table[Cos[t] X0 + Sin[t] Y0,
{t,-.5,0,.1}]
]}];
Clear[xyarc2]
xyarc2=Graphics3D[{color1,Line[
Table[Cos[t] X0 + Sin[t] Y0,
{t,polar[XY][[2]],polar[XY][[2]]+0.3,.1}]
]}];
Clear[stick]
stick=Graphics3D[{red,Line[{scale Z,XY}]}];

xaxislabel=Text["x",{0.363,0.26}];
yaxislabel = Text["y",{0.73, 0.30}];
zaxislabel = Text["z",{0.50, 0.67}];

stickplot =
Show[xyplane,axes,ray,point,Boxed->False,
ViewPoint->{2.747, 1.594, 1.168},
Epilog->{xaxislabel,yaxislabel,zaxislabel},
DisplayFunction->Identity];

rho=Text["ρ",{0.53, 0.45}];
rlabel = Show[xyplane,axes,ray,point,
Boxed->False,ViewPoint->{2.747, 1.594, 1.168},
Epilog->{rho,xaxislabel,yaxislabel,zaxislabel},
DisplayFunction->Identity];

phi=Text["ϕ",{0.55, 0.57}];
slabel =
Show[xyplane,zpoint,axes,ray,point,zarc,zarc1,zarc2,Boxed->False,ViewPoint->{2.747, 1.594, 1.168},
Epilog->{phi,xaxislabel,yaxislabel,zaxislabel},DisplayFunction->Identity];

theta=Text["θ",{0.5, 0.3}];

tlabel=Show[xyplane,axes,xypoint,xpoint,ray,stick,point,xyarc,xyarc1,xyarc2,Boxed->False,ViewPoint->{2.747, 1.594, 1.168},
Epilog->{theta,xaxislabel,yaxislabel,zaxislabel},
DisplayFunction->Identity];

all =
Show[xyplane,axes,ray,stick,point,zpoint,xpoint,xypoint,xyarc,xyarc1,xyarc2,zarc,zarc1,zarc2,Boxed->False,ViewPoint->{2.747, 1.594, 1.168},
Epilog->{theta,phi,rho,xaxislabel,yaxislabel,zaxislabel},DisplayFunction->Identity];

Cylindrical Setup.

Print["Cylindrical Setup"]
Clear[rrr, ttt, ppp1, ppp2, cylindricalsetup, ppppoint, pointlabel, 
      rrrlabel, tttlabel]
ttt = (π/4);
ppp1 = 
        ParametricPlot3D[{Cos[t], Sin[t], 0, {Red, Thickness[ .01]}}, {t, 0, 
            ttt}, Boxed -> False, Axes -> False, 
          DisplayFunction -> Identity];
ppp2 = 
        Graphics3D[{DarkGreen, Thickness[ .01], 
            Line[{{Cos[ttt], Sin[ttt], 0}, {Cos[ttt], Sin[ttt], 
                  1}}]}];
ppp3 = 
        Graphics3D[{Blue, Thickness[ .01], 
            Line[{{0, 0, 0}, {Cos[ttt], Sin[ttt], 0}}]}];
ppp4 = 
        Graphics3D[{Brown, PointSize[ .03], 
            Point[{Cos[ttt], Sin[ttt], 1}]}];
tttlabel = 
        Graphics3D[{Red, Text["θ", {1.4, 0.2, 0}]}];
rrrlabel = 
        Graphics3D[{Blue, Text["r", { .65, 0.15, 0}]}];
zzzlabel = 
        Graphics3D[{DarkGreen, Text["z", { .7,  .8, (1/2)}]}];
pointlabel = zzzlabel = 
          Graphics3D[{DarkGreen, 
              Text["{r Cos[θ],r Sin[θ], z}", {1.1, 1.1, 
                  1.3}]}];
cylindricalsetup = 
        Show[ppp1, ppp2, ppp3, ppp4, tttlabel, rrrlabel, zzzlabel, 
          pointlabel, ThreeAxes[1.4,  .2], 
          ViewPoint -> {3.149, -0.966, 0.774}, 
          PlotRange -> All, 
          DisplayFunction -> $DisplayFunction];

Crisscross

Clear[x]
crisscross = Plot[{Cos[x],Sin[x]},{x,Pi/8,3 Pi/8},
    PlotStyle->{{RGBColor[1,0,0],Thickness[0.02]}},
    Axes->None,AspectRatio->1/3,
    DisplayFunction->Identity];

Local Intializations

Clear[grad, expr, vars, point];
\(\(grad[expr_, \ vars_List, 
          point_List]\  := \ \(D\ [
expr, \ #] &\)\  /@ \ vars\ \  /. \ 
          Thread[vars\  -> point];\)

\)
grad[expr_, vars_] := grad[expr, vars, vars]
grad[expr_] := grad[expr, {x, y, z}, {x, y, z}]

\(
Clear[Curl]\)
\(Curl[expr_List, 
        vars_List] := \ 
{D[expr[\([3]\)], vars[\([2]\)]]\  - \ 
            D[expr[\([2]\)], vars[\([3]\)]], 
\ \(-\ 
              D[expr[\([3]\)], vars[\([1]\)]]\)\  + \ 
            D[expr[\([1]\)], vars[\([3]\)]], 
\ 
          D[expr[\([2]\)], vars[\([1]\)]]\  - \ 
            D[expr[\([1]\)], vars[\([2]\)]]} /; 
\ \(\(Length[expr] == 
          Length[vars]\)\(
\)\(
\)\)\)
Curl[expr_List] := Curl[expr, {x, y, z}];

\(
Clear[Div, expr, vars]\)
\(Div[expr_List, vars_List] := 
      Inner[D, expr, vars]\  /; \ 
\(\(Length[expr] == 
          Length[vars]\)\(
\)\(
\)\)\)
Div[expr_List] := Div[expr, {x, y, z}]

Clear[jacobian, jacobiandet]
\(jacobian[expr_List, vars_List] := 
      Outer[D, \ \ expr\ , 
          vars] /; 
\(\(Length[expr]\)\(==\)\(Length[
            vars]\)\(
\)\)\)
\(jacobiandet[expr_List, vars_List] := 

      Det[jacobian[expr, vars]] /; 
\(\(Length[expr] == 
          Length[vars]\)\(
\)\(\ \)\)\)

Clear[hessian, funs, vars]
\(\(hessian[funs_, \ vars_List, point_List]\  := \ 
        Outer[
D[funs, \ #1, \ #2] &, \ vars, \ vars] /. \ 

          Thread[vars -> point]\ ;\)\)
hessian[funs_,  vars_List] :=  
        hessian[funs, vars, vars];
hessian[funs_] := hessian[funs, {x, y}, {x, y}];
\(hessiandet[funs_, \ vars_List, point_List] := 
      Det[hessian[
funs, vars, point]]\)
\(hessiandet[funs_, \ vars_List] := 
      Det[hessian[
funs, vars, vars]]\)
hessiandet[funs_] := hessiandet[funs, {x, y}, {x, y}]

Clear[box]
\(box[
{x1_, x2_}, {y1_, y2_}, {z1_, z2_}, 

        color_:  RGBColor[1, 0, 0]] := 
      Module\ [
{side1, side2, side3, side4}, 


        side1 = Graphics3D[{color, Thickness[ .02], 

              Line[{{\ x1, y1, z1}, {x1, y2, z1}, 
\ \ \ \ \ \ {x1, y2, 
                    z2}, {x1, y1, z2}, {\ x1, y1, z1}}]}]; 


        side2 = Graphics3D[{color, Thickness[ .02], 

              Line[{{\ x1, y1, z1}, {x2, y1, z1}, 
\ \ \ \ \ \ {x2, y1, 
                    z2}, {x1, y1, z2}, {\ x1, y1, z1}}]}]; 


        side3 = Graphics3D[{color, Thickness[ .02], 

              Line[{{\ x2, y1, z1}, {x2, y2, z1}, 
\ \ \ \ \ \ {x2, y2, 
                    z2}, {x2, y1, z2}, {\ x2, y1, z1}}]}]; 
\ \ \ \ \ \ 

        side4 = Graphics3D[{color, Thickness[ .02], 

              Line[{{\ x1, y2, z1}, {x2, y2, z1}, 
\ \ \ \ \ \ {x2, y2, 
                    z2}, {x1, y2, z2}, {\ x1, y2, z1}}]}]; 
\ \ \ \ \ \ 
\ 
        Return[{side1, side2, side3, side4}]\ \ \ \ \ 

]


\)
Unprotect[SA];
Clear[SA, expr, vars];
\(\(SA[expr_List, vars_List] := 

        Norm[\ Cross[\ 
              D[expr, vars[\([1]\)]\ ]\ \ , 
\ \ \ \ \ \ \ \ \ \ \ \ \ D[\ 
                expr, vars[\([2]\)]]\ ]] /; 
\((Length[expr] == 3 && 
              Length[vars] == 2)\);\)
\)
\(\(Protect[SA];\)

\)
Unprotect[NV];
Clear[NV, expr, vars];
\(\(NV[expr_List, vars_List] := 

        Cross[\ D[expr, vars[\([1]\)]\ ]\ \ , 
\ \ \ \ \ \ \ \ \ \ \ \ \ D[\ 
              expr, vars[\([2]\)]]\ ] /; 
\((Length[expr] == 3 && 
              Length[vars] == 2)\);\)
\)
Protect[NV];

\(\(\( (*\(:\)\(Version : \ Mathematica\ 2.0\)\ *) \)\(
\)\)\)

◆ÁëëëáãÞ ìåôáâëçôþí óôï R\(TraditionalForm\`\^3\)

ÁëëáãÞ Ìáôáâëçôþí óå 3 äéáóôÜóåéò.

¸óôù x = g [u, v, w], y = h [u, v, w] and  z = k [u, v, w] Ýíáò 1-1 ìåôáó÷çìáôéóìüò åíüò ÷ùñßïõ G  óå uvw-óõíôåôáãìÝíåò åðß åíüò ÷ùñßïõ R óå xyz-óõíôåôáãìÝíåò.  ¸óôù óõíÜñôçóç f [x, y, z] ïñéóìÝíç åðß ôïõ R, Ýôóé þóôå g, h, k, êáé  f  íá Ý÷ïõí óõíå÷åßò ìåñéêÝò ðáñáãþãïõò êáé ç ÉáêùâéáíÞ 
J[u, v, w] = |({{∂(x/∂)u, ∂(x/∂)v, ∂(x/∂)w}, {∂(y/∂)u, ∂(y/∂)v, ∂(y/∂)w}, {∂(z/∂)u, ∂(z/∂)v, ∂(z/∂)w}})|   íá åßíáé ìç ìçäåíéêÞ åêôüò ßóùò óå ìåìïíùìÝíá óçìåßá. Ôüôå, 
	
∫∫∫\(TraditionalForm\`\+R\)  f [x, y, z] dx dy dz = 
∫∫∫\(TraditionalForm\`\+G\) f [g [u, v, w],h [u, v, w],k [u, v, w] ]  | J[u, v, w] | du dv dw.

◆ÊõëéíäñéêÝò ÓõíôåôáãìÝíåò

Ïé êõëéíäñéêÝò óõíôåôáãìÝíåò åßíáé ðïëý ÷ñÞóéìåò ãéá ôçí ðåñéãñáöÞ êõëéíäñéêþí óùìÜôùí êáé åðéöáíåéþí êáèþò êáé ðáñáâïëïåéäþí.

ÊõëéíäñéêÝò óõíôåôáãìÝíåò

Óôçí ðáñáêÜôù ãñáöéêÞ ðáñÜóôáóç ðåñéãñÜöïíôáé ïé êõëéíäñéêÝò óõíôåôáãìÝíåò.

Show[cylindricalsetup, 
        DisplayFunction -> $DisplayFunction];

Ïé êõëéíäñéêÝò óõíôåôáãìÝíåò åßíáé êáô' ïõóßáí ïé ðïëéêÝò óõíôåôáãìÝíåò óôï åðßðåäï xy ìáæß ìå ôçí ôñßôç óõíéóôþóá z. 
Óõíåðþò, óôéò êõëéíäñéêÝò óõíôåôáãìÝíåò åíüò óçìåßïõ, ç ôñßôç óõíéóôþóá åßíáé ôï ýøïò ôïõ óçìåßïõ áðü ôï åðßðåäï xy êáé ïé äýï ðñþôåò, r and θ, ðáñéóôÜíïõí áêñéâþò ôéò ðïëéêÝò óõíôåôáãìÝíåò ôçò ðñïâïëÞò ôïõ óçìåßïõ óôï åðßðåäï xy.

ÅÜí ãíùñßæïõìå ôéò êõëéíäñéêÝò óõíôåôáãìÝíåò r, θ, êáé z åíüò óçìåßïõ ôüôå ïé êáñôåóéáíÝò óõíôåôáãìÝíåò äßíïíôáé áðü ôéò ó÷Ýóåéò:

ÊáñôåóéáíÝò ùò óõíÜñôçóç êõëéíäñéêþí:
		x = r cos θ, 
		y = r sin è, êáé 
		z = z
Áíôßóôñïöá,

 ÊõëéíäñéêÝò ùò óõíÜñôçóç êáñôåóéáíþí:
 
		r = (Sqrt[(x^2) + (y^2)]), 
		θ = (({{arctan (y/x) , 
π + 
                  arctan (y/x) , , \(åÜí\ x > 0, \ y ≥ 0

                åÜí\ x < 0\)}, {2  π + 
                  arctan (y/x) ,, \(åÜí\ x > 0, \ y < 0\)}})), êáé 
		z = z

Ïé ðáñáêÜôù ó÷Ýóåéò ðåñéãñÜöïõí óå êõëéíäñéêÝò óõíôåôáãìÝíåò ôïí êýëéíäñï (ìå ôï åóùôåñéêü ôïõ) áêôßíáò  R áðü ôï åðßðåäï z = (Subscript[z, 1]) ìÝ÷ñé ôï åðßðåäï z = (Subscript[z, 2]) :
	0 ≤ r ≤ R,
	0 ≤ θ ≤ 2 π, êáé
	(Subscript[z, 1]) ≤ z ≤ (Subscript[z, 2])

Õðïëïãßæïõìå ôçí ïñßæïõóá ôçò ÉáêùâéáíÞò ôïõ ìåôáó÷çìáôéóìïý áðü êõëéíäñéêÝò óå êáñôåóéáíÝò óõíôåôáãìÝíåò :

Clear[x, y, z, r, s, θ, P, J, a, b, c]
P[r_, θ_, 
          z_] = {x[r_, θ_, z_], y[r_, θ_, z_], 
            z[r_, θ_, z_]} = {r Cos[θ], r Sin[θ], 
            z};
J[r_, θ_, z_] = 
      Simplify[jacobiandet[P[r, θ, z], {r, θ, z}]]

Óõíåðþò, ãéá ôéò êõëéíäñéêÝò óõíôåôáãìÝíåò èá ÷ñçóéìïðïéïýìå (óôï ðáñáðÜíù èÝùñçìá áëëáãÞò ìåôáâëçôþí) J[r,θ,z] = r. 
  
Õðåíèõìßæïõìå üôé áõôÞ ç ôéìÞ åßíáé ç ßäéá ìå ôçí ïñßæïõóá ôçò ÉáêùâéáíÞò ôïõ ìåôáó÷çìáôéóìïý ìå ôéò ðïëéêÝò óõíôåôáãìÝíåò óôï åðßðåäï.
   
ÓõìðåñáóìáôéêÜ, üôáí ïëïêëçñþíïõìå ìßá óõíÜñôçóç f åðß åíüò óôåñåïý R ôüôå, åÜí R* åßíáé ôï ßäéï óôåñåü åêðåöñáóìÝíï óå êõëéíäñéêÝò óõíôåôáãìÝíåò, éó÷ýåé  

∫∫∫\(TraditionalForm\`\+R\)  f (x, y, z) dx dy dz = ∫∫∫\(TraditionalForm\`\+\(R \( □\&*\)\)\) f (r cosè, r sinè, z) r dr dè dz.

ÃñáöéêÝò ðáñáóôÜóåéò ìå êõëéíäñéêÝò óõíôåôáãìÝíåò

ÐáñÜäåéãìá 1: Êýëéíäñïò

¸óôù ï êýëéíäñïò áêôßíáò 2 êáé ýøïõò 3 ï ïðïßïò óå êáñôåóéáíÝò óõíôåôáãìÝíåò äßíåôáé áðü ôéò ó÷Ýóåéò 
                   (x^2) + (y^2) = 4,  0≤z≤3.

Óå êõëéíäñéêÝò óõíôåôáãìÝíåò, ïé ðáñáðÜíù ó÷Ýóåéò ãßíïíôáé 
               r = 2, 0 ≤ è ≤ 2π êáé 0 ≤ z ≤ 3.
×ñçóéìïðïéþíôáò ôéò ôåëåõôáßåò ó÷Ýóåéò öôéÜ÷íïõìå åýêïëá ôçí ãñáöéêÞ ðáñÜóôáóç ôïõ êõëßíäñïõ.

Clear[x, y, z, r, s, θ, P, J, a, b, c, r0, z0]
P[r_, θ_, 
          z_] = {x[r_, θ_, z_], y[r_, θ_, z_], 
            z[r_, θ_, z_]} = {r Cos[θ], r Sin[θ], 
            z};
r0 = 2;
z0 = 3;
ParametricPlot3D[
        P[r0, θ, z], {θ, 0, 2 π}, {z, 0, 
          z0}];

ÐáñÜäåéãìá 2: ÔìÞìáôá êõëßíäñïõ

Ãéá ôçí ãñáöéêÞ ðáñÜóôáóç ôïõ ôìÞìáôïò ôïõ êõëßíäñïõ ãéá ôï ïðïßï éó÷ýåé  y≥0 ðåñéïñßæïõìå ôçí ãùíßá è ìå ôçí áíéóüôçôá 0≤è≤π.

Clear[x, y, z, r, s, θ, P, J, a, b, c, r0, z0]
P[r_, θ_, 
          z_] = {x[r_, θ_, z_], y[r_, θ_, z_], 
            z[r_, θ_, z_]} = {r Cos[θ], r Sin[θ], 
            z };
r0 = 2;
\(\(z0 = 3;\)
\)

ParametricPlot3D[
        P[r0, θ, z], {θ, 0,  π}, {z, 0, z0}, 
        AxesLabel -> {"x", "y", "z"}];

¼ìïéá ãéá x≤0.

ParametricPlot3D[
        P[r0, θ, z], {θ, π/2,  3  π/2}, {z, 0, z0}, 
        AxesLabel -> {"x", "y", "z"}];

ÐáñÜäåéãìá 3: ÐáñáâïëïåéäÝò

¸óôù ôï ðáñáâïëïåéäÝò z = (x^2) + (y^2),  0≤z≤25. Óå êõëéíäñéêÝò óõíôåôáãìÝíåò, ôï ðáñáâïëïåéäÝò áõôü áðïôåëåßôáé áðü üëá ôá óçìåßá (r,è,z) ôïõ ÷þñïõ ôá ïðïßá éêáíïðïéïýí ôéò ó÷Ýóåéò 
                0 ≤ r ≤ 5, 0≤ è ≤2π  êáé z =(r^2)
Ç ãñáöéêÞ ðáñÜóôáóç åßíáé

Clear[x, y, z, r, s, θ, P, J, a, b, c, r0, z0]
P[
          r_, θ_] = {x[r_, θ_], y[r_, θ_], 
            z[r_, θ_]} = {r Cos[θ], r Sin[θ], 
            r^2};
\(\(r0 = 5;\)
\)
ParametricPlot3D[
        P[r, θ], {θ, 0, 2 π}, {r, 0, r0}];

Ôï ðñþôï ôåôáñôéìüñéï ôïõ ðáñáâïëïåéäïýò z = (x^2) + (y^2),  0≤z≤9 äßíåôáé áðü ôéò ó÷Ýóåéò  
                0 ≤ r ≤ 3, 0≤ è ≤(π/2)  êáé z =(r^2)
Ç ãñáöéêÞ ðáñÜóôáóç åßíáé

Clear[x, y, z, r, s, θ, P, J, a, b, c, r0, z0]
P[
          r_, θ_] = {x[r_, θ_], y[r_, θ_], 
            z[r_, θ_]} = {r Cos[θ], r Sin[θ], 
            r^2};
\(\(r0 = 3;\)
\)
ParametricPlot3D[
        P[r, θ], {θ, 0, (π/2)}, {r, 0, r0}];

ÏëïêëÞñùóç ìå êõëéíäñéêÝò óõíôåôáãìÝíåò

ÐáñÜäåéãìá 4 :  Íá õðïëïãßóåôå ôï ïëïêëÞñùìá 

 ∫∫∫\(TraditionalForm\`\(\(\ \)\(\+R\)\)\) f(x,y,z)  \(\(\(dx\ dx\ dz\)\(\ \)\(,\)\)\(\ \ \ \ \)\(
\)\(\ \)\)  üðïõ  f(x,y,z) = (x^2) (y^2) (z^4) 
êáé R åßíáé ôï óôåñåü ðïõ öñÜóóåôáé áðü ôçí êõëéíäñéêÞ åðéöÜíåéá : (x^2) + (y^2) = 16,    0≤z≤7

Ç ðáñáìåôñéêïðïßçóç êáé ôï ãñÜöçìá ôçò ðáñáðÜíù åðéöÜíåéáò ÷ñçóéìïðïéþíôáò êõëéíäñéêÝò óõíôåôáãìÝíåò åßíáé:
 x = x [r,θ,z] = r Cos[θ]             0 ≤ r ≤ 4
 y = y [r,θ,z] = r Sin[θ]       ìå   0 ≤ θ ≤ 2π
 z = z [r,θ,z] = z                      0 ≤ z ≤ 7

Clear[P, r, θ, z, x, y, z]

\(\(P[r_, θ_, 
          z_] = \({x[r_, θ_, z_], y[r_, θ_, z_], 
            z[r_, θ_, z_]} = {r\ Cos[θ], r\ Sin[θ], 
            z}\);\)
\)
ParametricPlot3D[
        Evaluate[P[4, θ, z]], {θ, 0, 2 π}, {z, 0, 
          7}];

Õðïëïãßæïõìå ôï ïëïêëÞñùìá: Áðü ôï èåþñçìá áëëáãÞò ìåôáâëçôþí Ý÷ïõìå 

∫∫∫\(TraditionalForm\`\(\(\ \)\(\+R\)\)\)(x^2)(y^2)(z^4)\(\(dxdydz\)\(=\)\(
\)\(\ \)\)∫∫∫\(TraditionalForm\`\(\(\ \)\(\+\(R\^*\)\)\)\)\(TraditionalForm\`\(\((rCos[θ])\)\^2\) \(\((rSin[θ])\)\^2\) z\^4\)r \(\(dz\ dθ\ dr\)\(\ \)\(
\)\(\ \)\)
üðïõ ôï \(TraditionalForm\`\(R\^*\)\) åßíáé ôï ßäéï ÷ùñßï ìå ôï R åêðåöñáóìÝíï óå êõëéíäñéêÝò óõíôåôáãìÝíåò.
ÊÜíïõìå ôïí õðïëïãéóìü ìå ôï Mathematica :

Clear[J]

J[r_, θ_, z_] = 
      jacobiandet[P[r, θ, z], {r, θ, z}] // Simplify
f[x_, y_, z_] = (x^2) (y^2) (z^4);
(∫_0^4)(∫_0^(2 π))(∫_0^7) f[
              x[r, θ,  z], y[r, θ,  z], 
              z[r, θ,  z]] J[r, θ, 
              z] \[DifferentialD]z \[DifferentialD]θ \[DifferentialD]r

Ôé èá õðïëïãßæáìå åáí ç f Þôáí ç óôáèåñÞ óõíÜñôçóç  1 ?

Clear[f]
f[x_, y_, z_] = 1;
(∫_0^4)(∫_0^(2 π))(∫_0^7) f[
              x[r, θ,  z], y[r, θ,  z], 
              z[r, θ,  z]] J[r, θ, 
              z] \[DifferentialD]z \[DifferentialD]θ \[DifferentialD]r

¼ðùò âëÝðïõìå ôï ôåëåõôáßï áðïôÝëåóìá éóïýôáé ìå ôïí üãêï ôïõ êõëßíäñïõ ï ïðïßïò åßíáé  π(R^2) h=16π*7=112π.

ÐáñÜäåéãìá 5.  Íá âñåèåß ï üãêïò êõëßíäñïõ áêôßíáò R êáé ýøïõò h

×ñçóéìïðïéïýìå êáé ðÜëé êõëéíäñéêÝò óõíôåôáãìÝíåò.

Clear[x, y, z, r, P, J, R, h]
P[r_, θ_, 
          z_] = {x[r_, θ_, z_], y[r_, θ_, z_], 
            z[r_, θ_, z_]} = {r Cos[θ], r Sin[θ], 
            z};
J[r_, θ_, z_] = 
        Simplify[
          jacobiandet[P[r, θ, z], {r, θ, z}]];
f[x_, y_, z_] = 1;
(∫_0^R)(∫_0^(2 π))(∫_0^h) f[
              x[r, θ,  z], y[r, θ,  z], 
              z[r, θ,  z]] J[r, θ, 
              z] \[DifferentialD]z \[DifferentialD]θ \[DifferentialD]r

ÐáñÜäåéãìá 6.  Íá âñåèåß ï üãêïò ôïõ ðáñáâïëïåéäïýò (x^2) + (y^2) = 25 - z, 0≤z≤25.

Óå êõëéíäñéêÝò óõíôåôáãìÝíåò ôï ÷ùñßï ìáò ðåñéãñÜöåôáé áðü ôéò ó÷Ýóåéò 
   0 ≤ r ≤ 5,  
      0 ≤ è ≤ 2  π, 
      z = 25 - (r^2)  (0 ≤ z ≤ (r^2)).
   
 Ç ãñáöéêÞ ðáñÜóôáóç åßíáé :

Clear[x, y, z, r, s, θ, P, J, a, b, c, r0, z0]
P[
          r_, θ_] = {x[r_, θ_], y[r_, θ_], 
            z[r_, θ_]} = {r Cos[θ], r Sin[θ], 
            25 - r^2};
\(\(r0 = 5;\)
\)
ParametricPlot3D[
        P[r, θ], {θ, 0, 2 π}, {r, 0, r0}];

Clear[x, y, z, r, P, J, h]
P[r_, θ_, 
          z_] = {x[r_, θ_, z_], y[r_, θ_, z_], 
            z[r_, θ_, z_]} = {r Cos[θ], r Sin[θ], 
            z};
J[r_, θ_, z_] = 
        Simplify[
          jacobiandet[P[r, θ, z], {r, θ, z}]];
f[x_, y_, z_] = 1;
(∫_0^5)(∫_0^(2 π))(∫_0^((r^2)))f[x[r, θ,  z], y[r, θ,  z], z[r, θ,  z]] J[
              r, θ, 
              z] \[DifferentialD]z \[DifferentialD]θ \[DifferentialD]r

◆ÓöáéñéêÝò óõíôåôáãìÝíåò

Ïé óöáéñéêÝò óõíôåôáãìÝíåò åßíáé ðïëý ÷ñÞóéìåò ãéá ôçí ðåñéãñáöÞ óöáéñþí êáé êþíùí.

ÓöáéñéêÝò óõíôåôáãìÝíåò

Óôçí ðáñáêÜôù ãñáöéêÞ ðáñÜóôáóç ðåñéãñÜöïíôáé ïé óöáéñéêÝò óõíôåôáãìÝíåò :

Show[all, DisplayFunction -> $DisplayFunction];

Óôéò óöáéñéêÝò óõíôåôáãìÝíåò (ñ, ϕ, θ) åíüò óçìåßïõ Ñ, ôï ρ åßíáé ç áðüóôáóç ôïõ óçìåßïõ áðü ôçí áñ÷Þ ôùí áîüíùí, äçëáäÞ, 
                          ñ\(TraditionalForm\`\^2\) = (x^2)+ (y^2)+ (z^2) 
Ôï åõèýãñáììï ôìÞìá áðü ôï Ñ óôï 0 ó÷çìáôßæåé ìßá ãùíßá ìå ôïí èåôéêü Üîïíá  0z ç ïðïßá åßíáé ç äåýôåñç óõíôåôáãìÝíç ϕ.  ÐñïâÜëïíôáò áõôü ôï åõèýãñáììï ôìÞìá óôï åðßðåäï xy, ç ðñïâïëÞ ó÷çìáôßæåé ìéá ãùíßá ìå ôïí èåôéêü Üîïíá  0x ðïõ åßíáé ç ôñßôç óöáéñéêÞ óõíôåôáãìÝíç θ.  
Ç ôñéÜäá  (ρ, ϕ, θ) åßíáé ïé óöáéñéêÝò óõíôåôáãìÝíåò ôïõ Ñ êáé 
             ρ ϵ R+, ϕ ϵ [0,π] êáé  θ ϵ [0,2π].

Åáí ãíùñßæïõìå ôéò óöáéñéêÝò óõíôåôáãìÝíåò ρ, ϕ, êáé θ åíüò óçìåßïõ, ôüôå ïé êáñôåóéáíÝò óõíôåôáãìÝíåò ôïõ óçìåßïõ äßíïíôáé áðü ôéò åîÞò ó÷Ýóåéò (ëßãç Ôñéãùíïìåôñßá åßíáé áñêåôÞ):

ÊáñôåóéáíÝò óõíôåôáãìÝíåò ùò óõíÜñôçóç ôùí óöáéñéêþí :
		x = ρ sin ϕ cos θ, 
     		y = ρ sin ϕ sin θ, 
     		z = ρ cos ϕ.

Ç óöáßñá áêôßíáò R (ìáæß ìå ôï åóùôåñéêü ôçò) ðïõ ðåñéãñÜöåôáé áðü ôçí åîßóùóç  (x^2)+ (y^2)+ (z^2) ≤ (R^2),
äßíåôáé óå óöáéñéêÝò óõíôåôáãìÝíåò áðü ôéò åîÞò áíéóüôçôåò
	0 ≤ ρ ≤ R
	0 ≤ ϕ ≤ π êáé
	0 ≤ θ ≤ 2 π
Ç åðéöÜíåéá ôçò óöáßñáò áêôßíáò R äßíåôáé áðü ôçí åîßóùóç  ρ = R	.

Ðáñáäåßãìáôá ìå óöáéñéêÝò óõíôåôáãìÝíåò

ÐáñÜäåéãìá 1:  ÐïéÜ óçìåßá ôïõ (ℝ^3) Ý÷ïõí ϕ=π/2 ?

Ôá óçìåßá ôïõ xy-åðéðÝäïõ.

ÐáñÜäåéãìá 2:  ÐïéÜ óçìåßá ôïõ (ℝ^3) Ý÷ïõí θ=π ?

Ôï óçìåßá ðïõ áíÞêïõí óôï  xz- åðßðåäï êáé Ý÷ïõí  x < 0.

ÐáñÜäåéãìá 3:   ÐïéÜ óçìåßá ôïõ (ℝ^3) Ý÷ïõí ϕ=π ?

\(Ôá\ óçìåßá\ ôïõ\ \ z - Üîïíá\ \ ìå\ z < 0. \)

ÐáñÜäåéãìá 4:  ÐïéÜ óçìåßá ôïõ (ℝ^3)  éêáíïðïéïýí ôéò ó÷Ýóåéò ϕ=π/4 êáé θ=π/2 ?

\(TextForm\`Ç\ äéïxïôüìïò\ ôçò\ ãùíßáò\ \(\(yOz\)\(.\)\(\ \)\)\)

ÃñáöéêÝò ðáñáóôÜóåéò ìå óöáéñéêÝò óõíôåôáãìÝíåò

ÐáñÜäåéãìá 1:  ÃñáöéêÞ ðáñÜóôáóç óöáßñáò

Ç óöáßñá (x^2) + (y^2) + (z^2) = 9 ðåñéãñÜöåôáé óå óöáéñéêÝò óõíôåôáãìÝíåò ùò åîÞò: 
     ρ = 3,
     0 ≤ ϕ ≤ π, êáé  
     0 ≤ θ ≤ 2 π
Ìå ôçí ðáñáìåôñéêïðïßçóç, 

P(ϕ, θ) = (3Sin[ϕ] Cos[θ], 3Sin[ϕ] Sin[θ], 3Cos[ϕ])

ç ãñáöéêÞ ðáñÜóôáóç ìå ÷ñÞóç ôïõ Mathematica åßíáé

Clear[x, y, z, ρ, ϕ, θ, P]
P[ρ_, ϕ_, θ_] = {x[ρ_, ϕ_, θ_], 
            y[ρ_, ϕ_, θ_], 
            z[ρ_, ϕ_, θ_]} = {ρ Sin[ϕ] Cos[θ], ρ Sin[ϕ] Sin[θ], ρ Cos[ϕ]};
sphereplot = 
        ParametricPlot3D[
          P[3, ϕ, θ], {ϕ, 0, π}, {θ, 0, 
            2 π}, 
          AxesLabel -> {"x", "y", "z"}];

Ðñïöáíþò ç ðáñáìåôñéêïðïßçóç ôçò óöáßñáò ìå ÷ñÞóç êáñôåóéáíþí óõíôåôáãìÝíùí åßíáé ðïëý ðéï äýóêïëç.

ÐáñÜäåéãìá 2:  ÃñáöéêÞ ðáñÜóôáóç çìéóöáéñßùí êáé óöáéñéêþí ôìçìÜôùí

Ôï çìéóöáßñéï  (x^2) + (y^2) + (z^2) = 9, z≥0 ðåñéãñÜöåôáé óå óöáéñéêÝò óõíôåôáãìÝíåò áðü ôéò ó÷Ýóåéò
     ρ  =  3,
     0  ≤  ϕ  ≤  (π/2), and 
\(\(\ \ \ \ \ \)\(0\  ≤ \ θ\  ≤ \ \ 2  π\)\)
Ç ãñáöéêÞ ðáñÜóôáóç åßíáé

Clear[x, y, z, ρ, ϕ, θ, P]
P[ρ_, ϕ_, θ_] = {x[ρ_, ϕ_, θ_], 
            y[ρ_, ϕ_, θ_], 
            z[ρ_, ϕ_, θ_]} = {ρ Sin[ϕ] Cos[θ], ρ Sin[ϕ] Sin[θ], ρ Cos[ϕ]};
hemisphere = 
        ParametricPlot3D[
          P[3, ϕ, θ], {ϕ, 0, π/2}, {θ, 0, 
            2  π}, 
          AxesLabel -> {"x", "y", "z"}];

Ðåñéïñßæïíôáò ôçí ôñßôç (óöáéñéêÞ) óõíôåôáãìÝíç ϕ  áðü 0 Ýùò  π/4 ðáßñíïõìå Ýíá óöáéñéêü ôìÞìá

cap = 
        ParametricPlot3D[
          P[3, ϕ, θ], {ϕ, 0, π/4}, {θ, 0, 
            2  π}, 
          AxesLabel -> {"x", "y", "z"}];

Ãåíéêüôåñåò åðéëïãÝò óôá üñéá ôùí ϕ êáé θ äßíïõí ðéï ãåíéêÜ óöáéñéêÜ ôìÞìáôá, ãéá ðáñÜäåéãìá,

capg = 
        ParametricPlot3D[
          P[3, ϕ, θ], {ϕ, π/4, 
            2  π/3}, {θ, π/2, 7  π/4}, 
          AxesLabel -> {"x", "y", "z"}];

ÐáñÜäåéãìá 3:  ÃñáöéêÞ ðáñÜóôáóç êþíïõ

\(¸íáò\ êþíïò\ äßíåôáé\ óå\ êáñôåóéáíÝò\ óõíôåôáãìÝíåò, áðü\ ìßá\ åîßóùóç\ ôçò\ \(ìïñöÞò : \ 
\ \ \ \ \ \ \ \ \ \ \ \ \ z\) = \@\(x\^2 + y\^2\), ãéá\ 0 ≤ z ≤ 3. \)
(ç áíéóüôçôá ãéá ôï z Ýxåé óêïðü íá êÜíåé ôï xùñßï öñáãìÝíï)
\(Áöïý\ óå\ óöáéñéêÝò\ óõíôåôáãìÝíåò\ , \ \ \ \ \ \ \ 
\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ z = \(ρ\ cos\ ϕ\ \ êáé\ \ x\^2 + y\^2 = \(ρ\^2\) 
          sin\^2\ ϕ\), Ýðåôáé\ üôé\ ç\ åîßóùóç\ ôïõ\ êþíïõ\ óå\ óöáéñéêÝò\ óõíôåôáãìÝíåò\)
åßíáé 
                       (ρ^2) 
          (cos^2) ϕ  = (ρ^2) 
          (sin^2) ϕ
äçëáäÞ,  tan ϕ=1 Þ ϕ =π/4.

\(Óõíåðþò\ ï\ êþíïò\ \ \ z = \@\(x\^2 + y\^2\), 
        0 ≤ z ≤ 3\)  
\(óå\ óöáéñéêÝò\ óõíôåôáãìÝíåò\ ðåñéãñÜöåôáé\ áðü\ ôéò\ óxÝóåéò\) 
        0 ≤ ρ ≤ (3/(cos(( π)/4)))=3 (Sqrt[ 2])
        ϕ =(( π)/4), êáé
        0  ≤  θ  ≤   2  π

\(ÊÜíïõìå\ ôçí\ ãñáöéêÞ\ ðáñÜóôáóç\ ùò\ \(\(åîÞò\)\(:\)\)\)

Clear[x, y, z, ρ, ϕ, θ, P]
P[ρ_, ϕ_, θ_] = {x[ρ_, ϕ_, θ_], 
            y[ρ_, ϕ_, θ_], 
            z[ρ_, ϕ_, θ_]} = {ρ Sin[ϕ] Cos[θ], ρ Sin[ϕ] Sin[θ], ρ Cos[ϕ]};
coneplot = 
      ParametricPlot3D[
        P[ρ, π/4, θ], {ρ, 0, 3 (Sqrt[ 2])}, {θ, 0, 
          2  π}, Boxed -> False, 
        AxesLabel -> {"x", "y", "z"}]; 
    Show[coneplot, ViewPoint -> {4.164, 3.200, 1.031}]

ÐáñÜäåéãìá 4:  ÃñáöéêÞ ðáñÜóôáóç ðáãùôïý ÷ùíÜêé

\(TextForm\`¸íá\ ðáãùôü\ xùíÜêé\ ðåñéãñÜöåôáé\ óå\ êáñôåóéáíÝò\ \)
\(TextForm\`óõíôåôáãìÝíåò\ áðü\ \ áíéóüôçôåò\ ôçò\ \(ìïñöÞò\  : 

\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \@\(x\^2 + y\^2\) ≤ 
          z ≤ \@\(9 - x\^\(\(2\)\(\ \)\) - y\^2\)\)\)

ÊÜíïõìå ôçí ãñáöéêÞ ðáñÜóôáóç óå 2 âÞìáôá. Ðñþôá ôï ÷ùíÜêé:

Clear[x, y, z, ρ, ϕ, θ, cone]
x[ρ_, ϕ_, θ_]  =  ρ Sin[ϕ] Cos[θ];
y[ρ_, ϕ_, θ_]  =  ρ Sin[ϕ] Sin[θ];
\(\(z[ρ_, ϕ_, θ_] = \ ρ\ Cos[ϕ];\)
\)
cone[ρ_, θ_] = {x[ρ, (π/4), θ], 
          y[ρ, (π/4), θ], 
          z[ρ, (π/4), θ]};
coneplot = 
        ParametricPlot3D[
          cone[ρ, θ], {ρ, 0, 3}, {θ, 0, 2 π}, 
          Boxed -> False];

ÌåôÜ ôï ðáãùôü:

Clear[cap]
cap[ϕ_, θ_] = {x[3, ϕ, θ], 
          y[3, ϕ, θ], z[3, ϕ, θ]};
capplot = 
        ParametricPlot3D[
          cap[ϕ, θ], {ϕ, 0, (π/4)}, {θ, 0, 
            2 π}, Boxed -> False];

Êáé üëï ìáæß:

Show[capplot, coneplot, Boxed -> False];

ÂÝâáéá, ãéá íá ìïéÜæåé ç ãñáöéêÞ ðáñÜóôáóç ìå ðáãùôü ðñÝðåé íá ìåéþóïõìå ôçí óõíôåôáãìÝíç ϕ :

Clear[x, y, z, ρ, ϕ, θ, P]
P[ρ_, ϕ_, θ_] = {x[ρ_, ϕ_, θ_], 
            y[ρ_, ϕ_, θ_], 
            z[ρ_, ϕ_, θ_]} = {ρ Sin[ϕ] Cos[θ], ρ Sin[ϕ] Sin[θ], ρ Cos[ϕ]};
coneplot = 
        ParametricPlot3D[
          P[ρ, π/13, θ], {ρ, 0, (Sqrt[3])}, {θ, 0, 
            2  π}, Boxed -> False, 
          AxesLabel -> {"x", "y", "z"}];

ÏëïêëÞñùóç ìå óöáéñéêÝò óõíôåôáãìÝíåò

Ìåèïäïëïãßá êáé ðáñáäåßãìáôá

Õðïëïãßæïõìå ôçí ïñßæïõóá ôçò ÉáêùâéáíÞò ôïõ ìåôáó÷çìáôéóìïý áðü óöáéñéêÝò óå êáñôåóéáíÝò óõíôåôáãìÝíåò:

Clear[x, y, z, ρ, ϕ, θ, P, J]
P[ρ_, ϕ_, θ_] = {x[ρ_, ϕ_, θ_], 
            y[ρ_, ϕ_, θ_], 
            z[ρ_, ϕ_, θ_]} = {ρ Sin[ϕ] Cos[θ], ρ Sin[ϕ] Sin[θ], ρ Cos[ϕ]};
J[ρ_, ϕ_, θ_] = 
      jacobiandet[
        P[ρ, ϕ, θ], {ρ, ϕ, θ}]

Simplify[J[ρ, ϕ, θ]]

Óôéò óöáéñéêÝò óõíôåôáãìÝíåò Ý÷ïõìå  0 ≤ ϕ ≤ π.  Áöïý (ρ^(2 ))åßíáé ðÜíôá èåôéêü êáé sin ϕ  åßíáé ðÜíôá èåôéêü óôï äéÜóôçìá 
[0, π], ç ïñßæïõóá ôçò ÉáêùâéáíÞò äåí åßíáé ðïôÝ áñíçôéêÞ. ÊáôÜ óõíÝðåéá, ç áðüëõôç ôéìÞ ôçò ÉáêùâéáíÞò ïñßæïõóáò  |J[u, v, w]|, ðïõ åìöáíßæåôáé óôï Èåþñçìá áëëáãÞò ìåôáâëçôþí
∫∫∫\(TraditionalForm\`\+R\)  f [x, y, z] dx dy dz = 
∫∫∫\(TraditionalForm\`\+G\) f [g [u, v, w],h [u, v, w],k [u, v, w] ]  | J[u, v, w] | du dv dw.
åßíáé, óôçí ðåñßðôùóÞ ìáò |J[ρ, ϕ, θ] |=J[ρ, ϕ, θ] = (ρ^2) sin ϕ.

¼ôáí, óôï åîÞò, ïëïêëçñþíïõìå ìßá óõíÜñôçóç f åðß åíüò óôåñåïý R ôüôå, åÜí R* åßíáé ôï ßäéï óôåñåü åêðåöñáóìÝíï óå óöáéñéêÝò óõíôåôáãìÝíåò, èá éó÷ýåé  

∫∫∫\(TraditionalForm\`\+R\)f (x, y, z) dx dy dz =
       ∫∫∫\(TraditionalForm\`\+\(R\^*\)\)f (ρ sin ϕ cos è, ρ sin ϕ sin è, ρ  cos ϕ) (ρ^2) sin ϕ dρ dϕ dθ.

Ãíùóôïðïéïýìå êáé óôçí åöáñìïãÞ ìáò ôçí ôéìÞ ôçò  ôçò ÉáêùâéáíÞò ïñßæïõóáò ôïõ ìåôáó÷çìáôéóìïý áðü êáñôåóéáíÝò óå óöáéñéêÝò óõíôåôáãìÝíåò :

Clear[J]
J[ρ_, ϕ_, θ_] = (ρ^2) Sin[ϕ]

ÐáñÜäåéãìá  1:  ÏëïêëÞñùóç óõíÜñôçóçò åðß óöáßñáò (ìå ôï åóùôåñéêü ôçò)

Ìå ÷ñÞóç óöáéñéêþí óõíôåôáãìÝíùí õðïëïãßæïõìå ôï  
      ∫∫∫\(TraditionalForm\`\+S\)  (x^2)dz dx dy ,   üðïõ S :  (x^2) + (y^2) + (z^(2 ))≤ (a^2)

Clear[a, x, y, z, ρ, ϕ, θ, f, Vxyz]
f[x_, y_, z_] = (x^2)

Õðåíèõìßæïõìå üôé ç óöáßñá S  áêôßíáò a (ìå ôï åóùôåñéêü ôçò) ðåñéãñÜöåôáé áðü ôéò ðáñáêÜôù áíéóüôçôåò : 
   	0 ≤ ρ ≤ a,
	0 ≤ ϕ ≤ π, êáé
	0 ≤ θ ≤ 2 π

Óçìåßùóç:  Åáí èÝóïõìå  ρ = a, ôüôå ðáßñíïõìå ìüíï ôçí åðéöÜíåéá ôçò óöáßñáò. ÁëëÜ üôáí åðéôñÝðïõìå óôï ρ íá êéíåßôáé áðü 0 Ýùò a, ôüôå ðáßñíïõìå ïëüêëçñç ôçí óöáßñá ìå ôï åóùôåñéêü ôçò.

¸÷ïõìå, óýìöùíá ìå ôá ðáñáðÜíù,

∫∫∫\(TraditionalForm\`\+R\)  f[x,y,z]  dz dx dy = 
(∫_0^(2 π))(∫_0^π)(∫_0^a)  f[x[ρ, ϕ, θ], y[ρ, ϕ, θ], z[ρ, ϕ, θ] ] J[ρ, ϕ, θ] dρ dϕ dθ

Äçëþíïõìå ôçí áëëáãÞ ìåôáâëçôþí óôï Mathematica :

Clear[x, y, z, J, ρ, ϕ, θ]
x[ρ_, ϕ_, θ_]  =  ρ Sin[ϕ] Cos[θ]
y[ρ_, ϕ_, θ_]  =  ρ Sin[ϕ] Sin[θ]
z[ρ_, ϕ_, θ_] =  ρ Cos[ϕ]
J[ρ_, ϕ_, θ_] =  (ρ^2) Sin[ϕ]

Êáé õðïëïãßæïõìå ôï ïëïêëÞñùìá
(∫_0^(2 π))(∫_0^π)(∫_0^a) f (ρ sin ϕ cos è, ρ sin ϕ sin è, ρ  cos ϕ) (ρ^2) sin ϕ dρ dϕ dθ =
                = (∫_0^(2 π))(∫_0^π)(∫_0^a) (ρ^4) (sin^3) ϕ  (cos^2) è dρ dϕ dθ

Clear[a]
(∫_0^(2 π))(∫_0^π)(∫_0^a) 
              f[x[ρ, ϕ, θ], y[ρ, ϕ, θ], 
              z[ρ, ϕ, θ]] J[ρ, ϕ, θ] \[DifferentialD]ρ \[DifferentialD]ϕ \[DifferentialD]θ

ÐáñÜäåéãìá  2:  Õðïëïãéóìüò üãêïõ óöáßñáò áêôßíáò  a

Åáí ç óõíÜñôçóç  f(x,y,z) Þôáí ßóç ìå  1, ôï ðáñáðÜíù ïëïêëÞñùìá èá ìáò Ýäéíå ôïí üãêï ôçò óöáßñáò.

Clear[x, y, z, J, ρ, ϕ, θ, f]
f[x_, y_, z_] = 1;
x[ρ_, ϕ_, θ_]  =  ρ Sin[ϕ] Cos[θ];
y[ρ_, ϕ_, θ_]  =  ρ Sin[ϕ] Sin[θ];
z[ρ_, ϕ_, θ_] =  ρ Cos[ϕ];
J[ρ_, ϕ_, θ_] = 
        Simplify[
          jacobiandet[
            P[ρ, ϕ, θ], {ρ, ϕ, θ}]];
Clear[a]
(∫_0^(2 π))(∫_0^π)(∫_0^a) 
              f[x[ρ, ϕ, θ], y[ρ, ϕ, θ], 
              z[ρ, ϕ, θ]] J[ρ, ϕ, θ] \[DifferentialD]ρ \[DifferentialD]ϕ \[DifferentialD]θ

Áõôü âåâáßùò ôï îÝñáìå ðñéí ìÜèïõìå ôñéðëÜ ïëïêëçñþìáôá !  (Quiz: ðùò ôï îÝñáìå ?)

ÐáñÜäåéãìá  3:  Õðïëïãéóìüò üãêïõ åëëåéøïåéäïýò 
                          ((x^2)/(a^2)) +((y^2)/(b^2)) + ((z^2)/(c^2)) ≤1

Ãíùñßæïíôáò ôï ôñüðï ìå ôïí ïðïßï ç Ýëëåéøç êáé ï êýêëïò ó÷åôßæïíôáé óôéò 2 äéáóôÜóåéò åßíáé ëïãéêü íá ðåñéìÝíïõìå áíÜëïãç ó÷Ýóç êáé óôéò 3 äéáóôÜóåéò. ÐñÜãìáôé, ç ðáñáìåôñéêïðïßçóç ôïõ åëëåéøïåéäïýò ðïõ ðåñéãñÜöåôáé áðü ôçí áíßóùóç
                          ((x^2)/(a^2)) +((y^2)/(b^2)) + ((z^2)/(c^2)) ≤1 
åðéôõã÷Üíåôáé ìå ìéêñÞ ôñïðïðïßçóç ôùí óöáéñéêþí óõíôåôáãìÝíùí ùò åîÞò:

Clear[x, y, z, ρ, ϕ, θ, P, a, b, c, J]
P[ρ_, ϕ_, θ_] = {x[ρ_, ϕ_, θ_], 
            y[ρ_, ϕ_, θ_], 
            z[ρ_, ϕ_, θ_]} = {a ρ Sin[ϕ] Cos[θ], b ρ Sin[ϕ] Sin[θ], 
            c ρ Cos[ϕ]};

ÐñÜãìáôé,  áíôéêáèéóôþíôáò ôéò ðáñáðÜíù ðáñáìåôñéêÝò åîéóþóåéò óôçí ðáñÜóôáóç ((x^2)/(a^2)) +((y^2)/(b^2)) + ((z^2)/(c^2)) Ý÷ïõìå:

\(Simplify[
      x[ρ, ϕ, θ]\^2\/a\^2\  + \ 
        y[ρ, ϕ, θ]\^2\/b\^2 + \ 
        z[ρ, ϕ, θ]\^2\/c\^2]\)

Ãéá íá ðÜñïõìå ôçí åðéöÜíåéá ôïõ åëëåéøïåéäïýò, äçë. ôá óçìåßá ðïõ éêáíïðïéïýí ôçí éóüôçôçá ((x^2)/(a^2)) +((y^2)/(b^2)) + ((z^2)/(c^2)) =1 áñêåß íá èÝóïõìå ρ = 1.  ÄçëáäÞ,

Clear[x, y, z, ρ, ϕ, θ, ellipsoid]
x[ρ_, ϕ_, θ_]  =  
        a ρ Sin[ϕ] Cos[θ];
y[ρ_, ϕ_, θ_]  =  
        b ρ Sin[ϕ] Sin[θ];
z[ρ_, ϕ_, θ_]  =  
        c ρ Cos[ϕ];
a = 3;
b = 4;
c = 5;
ellipsoid[ϕ_, θ_] = {x[1, ϕ, θ], 
          y[1, ϕ, θ], z[1, ϕ, θ]};
ellipsoidplot = 
        ParametricPlot3D[
          ellipsoid[ϕ, θ], {ϕ, 0, π}, {θ, 0, 
            2 π}, Boxed -> False];

Õðïëïãßæïõìå ôçí ïñßæïõóá ôçò ÉáêùâéáíÞò ôïõ ðáñáðÜíù ìåôáó÷çìáôéóìïý

Clear[x, y, z, ρ, ϕ, θ, P, a, b, c, J]
P[ρ_, ϕ_, θ_] = {x[ρ_, ϕ_, θ_], 
            y[ρ_, ϕ_, θ_], 
            z[ρ_, ϕ_, θ_]} = {a ρ Sin[ϕ] Cos[θ], b ρ Sin[ϕ] Sin[θ], 
            c ρ Cos[ϕ]};
J[ρ_, ϕ_, θ_] = 
      jacobiandet[P[ρ, ϕ, θ], {ρ, ϕ, θ}] // 
        Simplify

Ðñïöáíþò, ôï áðïôÝëåóìá åßíáé ðáñåìöåñÝò ìå åêåßíï ôïõ ìåôáó÷çìáôéóìïý ãéá ôéò êáíïíéêÝò óöáéñéêÝò óõíôåôáãìÝíåò. 
¸÷ïíôáò êÜíåé ôçí ðáñáðÜíù áëëáãÞ ìåôáâëçôþí, ç åýñåóç ôïõ üãêïõ ôïõ åëëåéøïåéäïýò åßíáé ðáíåýêïëç

(∫_0^(2 π))(∫_0^π)(∫_0^1) 
            1  J[ρ, ϕ, θ] \[DifferentialD]ρ \[DifferentialD]ϕ \[DifferentialD]θ

ÐáñÜäåéãìá  4:  Õðïëïãéóìüò üãêïõ ðáãùôïý ÷ùíÜêé

¼ôáí êÜíáìå ôéò ãñáöéêÝò ðáñáóôÜóåéò ðáñáðÜíù, ðáñáôçñïýìå üôé ãéá ôïí êþíï èåùñÞóáìå ôçí ðáñáìåôñéêïðïßçóç 

P(ñ,θ)={x[ñ,ϕ,θ],y[ñ,ϕ,θ],z[ñ,ϕ,θ]}    
           =  {ñ Sin[(π/4)] Cos[θ],ñ Sin[(π/4)] Sin[θ],ñ Cos[(π/4)]}

ìå 0 ≤ ρ ≤ 3, êáé 0 ≤ θ ≤ 2π.

Ãéá ôï ðáãùôü èåùñÞóáìå ôçí ðáñáìåôñéêïðïßçóç 

P(ϕ, θ)={x[
            3, ϕ, θ], y[3, ϕ, θ], 
          z[3, ϕ, θ]}  
           = {3 Sin[ϕ] Cos[θ], 
        3  Sin[ϕ] Sin[θ], 3  Cos[ϕ]}

ìå 0 ≤ϕ ≤ (π/4), êáé 0 ≤ θ ≤ 2π.
Óõíåðþò, èåùñþíôáò 
                        0 ≤ ρ ≤ 3, 
                        0 ≤ ϕ ≤ (π/4), êáé
                        0 ≤ θ ≤ 2π,
ôá óçìåßá 

        {x[ρ, ϕ, θ], y[ρ, ϕ, θ], 
        z[ρ, ϕ, θ]} 

èá "óáñþóïõí" üëï ôï óôåñåü Å (=ðáãùôü ÷ùíÜêé). 
Ãéá íá âñïýìå ôïí üãêï ôïõ Å ôï ìüíï ðïõ ÷ñåéÜæåôáé íá êÜíïõìå åßíáé íá õðïëïãßóïõìå ôï ðáñáêÜôù ïëïêëÞñùìá 

   ¼ãêïò Å = ∫∫∫\(TraditionalForm\`\+R\) 1 dz dx dy 
                                   = (∫_0^(2 π))(∫_0^((π/4)))(∫_0^3)  (ñ^2) Sin[ϕ] dρ dϕ dθ

Clear[x, y, z, ρ, ϕ, θ, P, J]
P[ρ_, ϕ_, θ_] = {x[ρ_, ϕ_, θ_], 
            y[ρ_, ϕ_, θ_], 
            z[ρ_, ϕ_, θ_]} = {ρ Sin[ϕ] Cos[θ], ρ Sin[ϕ] Sin[θ], ρ Cos[ϕ]};
J[ρ_, ϕ_, θ_] = 
      jacobiandet[P[ρ, ϕ, θ], {ρ, ϕ, θ}] // 
        Simplify
ans = (∫_0^(2 π))(∫_0^((π/4)))(∫_0^3) 
            J[ρ, ϕ, θ] \[DifferentialD]ρ \[DifferentialD]ϕ \[DifferentialD]θ

Êáé ãéá íá ðÜñïõìå ìßá éäÝá ôïõ ðüóï ðáãùôü ôñþìå :

N[ans]