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-{
- "metadata": {
- "name": "chapter_2.ipynb"
- },
- "nbformat": 3,
- "nbformat_minor": 0,
- "worksheets": [
- {
- "cells": [
- {
- "cell_type": "markdown",
- "metadata": {},
- "source": [
- "<h1>Chapter 2: Coordinate Systems and Transformation<h1>"
- ]
- },
- {
- "cell_type": "markdown",
- "metadata": {},
- "source": [
- "<h3>Example 2.1, Page number: 36<h3>"
- ]
- },
- {
- "cell_type": "code",
- "collapsed": false,
- "input": [
- "'''\n",
- "Given point P(-2,6,3) and vector A = y a_x + (x+z) a_y ,\n",
- "express P and A in cylindrical and spherical coordinates.\n",
- "Evaluate A at P in the Cartesian, cylindrical, and spherical systems. '''\n",
- "\n",
- "import scipy\n",
- "from numpy import *\n",
- "\n",
- "#Variable Declaration\n",
- "\n",
- "x=-2\n",
- "y=6\n",
- "z=3 \n",
- "\n",
- "#Calculations\n",
- "\n",
- "r=scipy.sqrt(x**2+y**2)\n",
- "phi=scipy.arctan(-y/x) #phi in radians in 1st quadrant\n",
- "phid=180-(phi*180/scipy.pi) #phi in degrees in second quadrant\n",
- "phic=scipy.pi*phid/180.0 #phi in radians in second quadrant\n",
- "R=scipy.sqrt(x**2+y**2+z**2) \n",
- "theta=scipy.arctan(r/z) \n",
- "\n",
- " #P in cylindrical coordinates\n",
- " \n",
- "Pcyl=array([round(r,2),round(phid,2),z]) \n",
- "\n",
- " #P in spherical coordinates\n",
- " \n",
- "Psph=array([round(R,2),round(theta*180/scipy.pi,2),\n",
- " round(phid,2)]) \n",
- "\n",
- " #Vector A in cylindrical coordinate system\n",
- "\n",
- "Xc=r*scipy.cos(phic)\n",
- "Yc=r*scipy.sin(phic)\n",
- "Zc=z \n",
- "Ar=Yc*scipy.cos(phic)+(Xc+Zc)*scipy.sin(phic)\n",
- "Aphi=-Yc*scipy.sin(phic)+(Xc+Zc)*scipy.cos(phic)\n",
- "Az=0\n",
- "Acyl=array([round(Ar,4),round(Aphi,3),Az])\n",
- "\n",
- " #Vector A in spherical coordinate system\n",
- "\n",
- "Xs=R*scipy.cos(phic)*scipy.sin(theta)\n",
- "Ys=R*scipy.sin(phic)*scipy.sin(theta)\n",
- "Zs=R*scipy.cos(theta)\n",
- "AR=Ys*scipy.sin(theta)*scipy.cos(phic)+(Xs+Zs)*scipy.sin(theta)*scipy.sin(phic) \n",
- "Ath=Ys*scipy.cos(theta)*scipy.cos(phic)+(Xs+Zs)*scipy.cos(theta)*scipy.sin(phic) \n",
- "Aph=-Ys*scipy.sin(phic)+(Xs+Zs)*scipy.cos(phic)\n",
- "Asph=array([round(AR,4),round(Ath,4),round(Aph,3)])\n",
- "\n",
- "#Results\n",
- "\n",
- "print 'P in cylindrical coordinates =',Pcyl\n",
- "print 'P in spherical coordinates =',Psph\n",
- "print 'A in cylindrical coordinates =',Acyl\n",
- "print 'A in spherical coordinates =',Asph"
- ],
- "language": "python",
- "metadata": {},
- "outputs": [
- {
- "output_type": "stream",
- "stream": "stdout",
- "text": [
- "P in cylindrical coordinates = [ 6.32 108.43 3. ]\n",
- "P in spherical coordinates = [ 7. 64.62 108.43]\n",
- "A in cylindrical coordinates = [-0.9487 -6.008 0. ]\n",
- "A in spherical coordinates = [-0.8571 -0.4066 -6.008 ]\n"
- ]
- }
- ],
- "prompt_number": 1
- },
- {
- "cell_type": "markdown",
- "metadata": {},
- "source": [
- "<h3>Example 2.2, Page number: 39<h3>"
- ]
- },
- {
- "cell_type": "code",
- "collapsed": false,
- "input": [
- "'''\n",
- "Express vector B = 10/R a_r+ r cos(theta) a_(theta) + a(phi)\n",
- "in Cartesian and cylindrical coordinates. \n",
- "Find B (- 3, 4, 0) and B (5, pi/2, - 2). '''\n",
- "\n",
- "import scipy\n",
- "from numpy import *\n",
- " \n",
- "#Variable Declaration\n",
- "\n",
- "x=-3\n",
- "y=4\n",
- "z=0\n",
- "p=5\n",
- "phi=scipy.pi/2 \n",
- "Zc=-2\n",
- "\n",
- "#Calculations\n",
- "\n",
- " #B in cartesian coordinates\n",
- "\n",
- "R=scipy.sqrt(x**2+y**2+z**2)\n",
- "r=scipy.sqrt(x**2+y**2) \n",
- "P=scipy.arcsin(r/R) #in radians\n",
- "Q=scipy.arccos(x/r) #in radians \n",
- "f=10/R \n",
- "Bx=f*scipy.sin(P)*scipy.cos(Q)+R*(scipy.cos(P))**2*scipy.cos(Q)-scipy.sin(Q) \n",
- "By=f*scipy.sin(P)*scipy.sin(Q)+R*(scipy.cos(P))**2*scipy.sin(Q)+scipy.cos(Q) \n",
- "Bz=f*scipy.cos(P)-R*scipy.cos(P)*scipy.sin(P) \n",
- "Bcart=array([round(Bx,0),round(By,0),round(-Bz,0)])\n",
- "\n",
- " #B in cylindrical coordinates\n",
- " \n",
- "Rc=sqrt(p**2+Zc**2) \n",
- "Pc=scipy.arccos(Zc/Rc) #in radians\n",
- "Br=(10/Rc)*scipy.sin(Pc)+Rc*(scipy.cos(Pc))**2 \n",
- "Bp=1 \n",
- "Bzc=(10/Rc)*scipy.cos(Pc)-Rc*scipy.cos(Pc)*scipy.sin(Pc) \n",
- "Bcyl=array([round(Br,3),Bp,round(Bzc,3)])\n",
- "\n",
- "#Results\n",
- "\n",
- "print 'B(-3,4,0) in cartesian coordinates is',Bcart\n",
- "print 'B(5,pi/2,-2) in cylindrical coordinates is',Bcyl"
- ],
- "language": "python",
- "metadata": {},
- "outputs": [
- {
- "output_type": "stream",
- "stream": "stdout",
- "text": [
- "B(-3,4,0) in cartesian coordinates is [-2. 1. 0.]\n",
- "B(5,pi/2,-2) in cylindrical coordinates is [ 2.467 1. 1.167]\n"
- ]
- }
- ],
- "prompt_number": 2
- },
- {
- "cell_type": "markdown",
- "metadata": {},
- "source": [
- "<h3>Example 2.3, Page number: 44<h3>"
- ]
- },
- {
- "cell_type": "code",
- "collapsed": false,
- "input": [
- "'''\n",
- "Two uniform vector fields are gIven by E=-5a_p+10a_phi+3a_z and\n",
- "F=a_p+2a_phi-6a_z. Calculate \n",
- "(a) |E X F|\n",
- "(b) The vector component of E at P(5, pi/2, 3) parallel to the line x=2, z=3 \n",
- "(c) The angle E makes with the surface z = 3 at P '''\n",
- "\n",
- "import scipy\n",
- "from numpy import *\n",
- "\n",
- "#Variable Declaration\n",
- "\n",
- "E=array([-5,10,3]) #in cylindrical coordinates\n",
- "F=array([1,2,-6]) #in cylindrical coordinates\n",
- "P=array([5,scipy.pi/2,3]) #in cylindrical coordinates\n",
- "\n",
- "#Calculations\n",
- "\n",
- "exf=cross(E,F)\n",
- "ansa=scipy.sqrt(dot(exf,exf)) #|EXF|\n",
- "ay=array([round(scipy.sin(scipy.pi/2),0),\n",
- " round(scipy.cos(scipy.pi/2),0),0])\n",
- "ansb=dot(E,ay)*ay\n",
- "modE=scipy.sqrt(dot(E,E))\n",
- "az=array([0,0,1])\n",
- "thetaEz=(180/scipy.pi)*arccos(dot(E,az)/modE) #in degrees\n",
- "ansc=90-thetaEz #in degrees\n",
- "\n",
- "#Results\n",
- "\n",
- "print '|EXF| =',round(ansa,2)\n",
- "print 'The vector component of E at P parallel to the line x=2,z=3 =',ansb,','\n",
- "print 'in spherical coordinates'\n",
- "print 'The angle E makes with the surface z = 3 at P =',round(ansc,2),'degrees'"
- ],
- "language": "python",
- "metadata": {},
- "outputs": [
- {
- "output_type": "stream",
- "stream": "stdout",
- "text": [
- "|EXF| = 74.06\n",
- "The vector component of E at P parallel to the line x=2,z=3 = [-5. -0. -0.] ,\n",
- "in spherical coordinates\n",
- "The angle E makes with the surface z = 3 at P = 15.02 degrees\n"
- ]
- }
- ],
- "prompt_number": 3
- },
- {
- "cell_type": "markdown",
- "metadata": {},
- "source": [
- "<h3>Example 2.4, Page number: 45<h3>"
- ]
- },
- {
- "cell_type": "code",
- "collapsed": false,
- "input": [
- " '''\n",
- "Given a vector field \n",
- "D = r*sin(phi)a_r-(1/r)*sin(theta)*cos(phi)a_theta+ r^2*a_phi, determine\n",
- "(a) D at P(10,150\u00b0,330\u00b0) \n",
- "(b) The component of D tangential to the spherical surface r = 10 at P \n",
- "(c) A unit vector at P perpendicular to D and tangential to the cone 0 = 150\u00b0 '''\n",
- "\n",
- "import scipy\n",
- "from numpy import *\n",
- "\n",
- "#Variable Declaration\n",
- "\n",
- "aR=array([1,0,0]) #Unit vector along radial direction\n",
- "ath=array([0,1,0]) #Unit vector along theta direction\n",
- "aph=array([0,0,1]) #Unit vector along phi direction\n",
- "P=array([10,scipy.pi*150/180,scipy.pi*330/180])\n",
- "\n",
- "#Calculations\n",
- "\n",
- "r=dot(P,aR)\n",
- "q=dot(P,aph)\n",
- "p=dot(P,ath)\n",
- "R=r*scipy.sin(q)\n",
- "Ph=-scipy.sin(p)*scipy.cos(q)/r\n",
- "Q=r*r\n",
- "D=array([R,Ph,Q]) #D at P(10,150\u00b0,330\u00b0)\n",
- "rDr=round(dot(aR,D),0) #radial component of D\n",
- "rDth=round(dot(-ath,D),3) #theta component of D\n",
- "rDph=round(dot(aph,D),0) #phi component of D\n",
- "\n",
- "Dn=array([r*scipy.sin(q),0,0]) #Component of D normal to surface r=10\n",
- "Dt=D-Dn #Component of D tangential to surface r=10\n",
- "Dtr=round(dot(aR,Dt),0) #radial component of Dt\n",
- "Dtth=round(dot(-ath,Dt),3) #theta component of Dt\n",
- "Dtph=round(dot(aph,Dt),0) #phi component of Dt\n",
- "rDt=array([Dtr,Dtth,Dtph])\n",
- "\n",
- " #Unit vector normal to D and tangential to cone theta=45 degrees\n",
- "\n",
- "U=cross(D,ath)\n",
- "u=U/scipy.sqrt(dot(U,U)) \n",
- "ru=array([round(dot(aR,u),4),round(dot(ath,u),4),round(dot(aph,u),4)])\n",
- "\n",
- "#Results\n",
- "\n",
- "print 'D at P(10,150\u00b0,330\u00b0) = [',rDr,' ',rDth,' ',rDph,']'\n",
- "print 'The component of D tangential to the spherical surface r = 10 at P ='\n",
- "print '[',Dtr,' ',Dtth,' ',Dtph,']'\n",
- "print 'A unit vector at P perpendicular to D and tangential to cone 0 = 150\u00b0 ='\n",
- "print ru"
- ],
- "language": "python",
- "metadata": {},
- "outputs": [
- {
- "output_type": "stream",
- "stream": "stdout",
- "text": [
- "D at P(10,150\u00b0,330\u00b0) = [ -5.0 0.043 100.0 ]\n",
- "The component of D tangential to the spherical surface r = 10 at P =\n",
- "[ 0.0 0.043 100.0 ]\n",
- "A unit vector at P perpendicular to D and tangential to cone 0 = 150\u00b0 =\n",
- "[-0.9988 0. -0.0499]\n"
- ]
- }
- ],
- "prompt_number": 10
- }
- ],
- "metadata": {}
- }
- ]
+{
+ "metadata": {
+ "name": "",
+ "signature": "sha256:d78c85d754b20d2817dcaff01d1e4d9adbe09676da8e1b1ab97dcc0ae047a88a"
+ },
+ "nbformat": 3,
+ "nbformat_minor": 0,
+ "worksheets": [
+ {
+ "cells": [
+ {
+ "cell_type": "markdown",
+ "metadata": {},
+ "source": [
+ "<h1>Chapter 2: Coordinate Systems and Transformation<h1>"
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "metadata": {},
+ "source": [
+ "<h3>Example 2.1, Page number: 36<h3>"
+ ]
+ },
+ {
+ "cell_type": "code",
+ "collapsed": false,
+ "input": [
+ " \n",
+ "import scipy\n",
+ "from numpy import *\n",
+ "\n",
+ "#Variable Declaration\n",
+ "\n",
+ "x=-2\n",
+ "y=6\n",
+ "z=3 \n",
+ "\n",
+ "#Calculations\n",
+ "\n",
+ "r=scipy.sqrt(x**2+y**2)\n",
+ "phi=scipy.arctan(-y/x) #phi in radians in 1st quadrant\n",
+ "phid=180-(phi*180/scipy.pi) #phi in degrees in second quadrant\n",
+ "phic=scipy.pi*phid/180.0 #phi in radians in second quadrant\n",
+ "R=scipy.sqrt(x**2+y**2+z**2) \n",
+ "theta=scipy.arctan(r/z) \n",
+ "\n",
+ " #P in cylindrical coordinates\n",
+ " \n",
+ "Pcyl=array([round(r,2),round(phid,2),z]) \n",
+ "\n",
+ " #P in spherical coordinates\n",
+ " \n",
+ "Psph=array([round(R,2),round(theta*180/scipy.pi,2),\n",
+ " round(phid,2)]) \n",
+ "\n",
+ " #Vector A in cylindrical coordinate system\n",
+ "\n",
+ "Xc=r*scipy.cos(phic)\n",
+ "Yc=r*scipy.sin(phic)\n",
+ "Zc=z \n",
+ "Ar=Yc*scipy.cos(phic)+(Xc+Zc)*scipy.sin(phic)\n",
+ "Aphi=-Yc*scipy.sin(phic)+(Xc+Zc)*scipy.cos(phic)\n",
+ "Az=0\n",
+ "Acyl=array([round(Ar,4),round(Aphi,3),Az])\n",
+ "\n",
+ " #Vector A in spherical coordinate system\n",
+ "\n",
+ "Xs=R*scipy.cos(phic)*scipy.sin(theta)\n",
+ "Ys=R*scipy.sin(phic)*scipy.sin(theta)\n",
+ "Zs=R*scipy.cos(theta)\n",
+ "AR=Ys*scipy.sin(theta)*scipy.cos(phic)+(Xs+Zs)*scipy.sin(theta)*scipy.sin(phic) \n",
+ "Ath=Ys*scipy.cos(theta)*scipy.cos(phic)+(Xs+Zs)*scipy.cos(theta)*scipy.sin(phic) \n",
+ "Aph=-Ys*scipy.sin(phic)+(Xs+Zs)*scipy.cos(phic)\n",
+ "Asph=array([round(AR,4),round(Ath,4),round(Aph,3)])\n",
+ "\n",
+ "#Results\n",
+ "\n",
+ "print 'P in cylindrical coordinates =',Pcyl\n",
+ "print 'P in spherical coordinates =',Psph\n",
+ "print 'A in cylindrical coordinates =',Acyl\n",
+ "print 'A in spherical coordinates =',Asph"
+ ],
+ "language": "python",
+ "metadata": {},
+ "outputs": [
+ {
+ "output_type": "stream",
+ "stream": "stdout",
+ "text": [
+ "P in cylindrical coordinates = [ 6.32 108.43 3. ]\n",
+ "P in spherical coordinates = [ 7. 64.62 108.43]\n",
+ "A in cylindrical coordinates = [-0.9487 -6.008 0. ]\n",
+ "A in spherical coordinates = [-0.8571 -0.4066 -6.008 ]\n"
+ ]
+ }
+ ],
+ "prompt_number": 1
+ },
+ {
+ "cell_type": "markdown",
+ "metadata": {},
+ "source": [
+ "<h3>Example 2.2, Page number: 39<h3>"
+ ]
+ },
+ {
+ "cell_type": "code",
+ "collapsed": false,
+ "input": [
+ " \n",
+ "import scipy\n",
+ "from numpy import *\n",
+ " \n",
+ "#Variable Declaration\n",
+ "\n",
+ "x=-3\n",
+ "y=4\n",
+ "z=0\n",
+ "p=5\n",
+ "phi=scipy.pi/2 \n",
+ "Zc=-2\n",
+ "\n",
+ "#Calculations\n",
+ "\n",
+ " #B in cartesian coordinates\n",
+ "\n",
+ "R=scipy.sqrt(x**2+y**2+z**2)\n",
+ "r=scipy.sqrt(x**2+y**2) \n",
+ "P=scipy.arcsin(r/R) #in radians\n",
+ "Q=scipy.arccos(x/r) #in radians \n",
+ "f=10/R \n",
+ "Bx=f*scipy.sin(P)*scipy.cos(Q)+R*(scipy.cos(P))**2*scipy.cos(Q)-scipy.sin(Q) \n",
+ "By=f*scipy.sin(P)*scipy.sin(Q)+R*(scipy.cos(P))**2*scipy.sin(Q)+scipy.cos(Q) \n",
+ "Bz=f*scipy.cos(P)-R*scipy.cos(P)*scipy.sin(P) \n",
+ "Bcart=array([round(Bx,0),round(By,0),round(-Bz,0)])\n",
+ "\n",
+ " #B in cylindrical coordinates\n",
+ " \n",
+ "Rc=sqrt(p**2+Zc**2) \n",
+ "Pc=scipy.arccos(Zc/Rc) #in radians\n",
+ "Br=(10/Rc)*scipy.sin(Pc)+Rc*(scipy.cos(Pc))**2 \n",
+ "Bp=1 \n",
+ "Bzc=(10/Rc)*scipy.cos(Pc)-Rc*scipy.cos(Pc)*scipy.sin(Pc) \n",
+ "Bcyl=array([round(Br,3),Bp,round(Bzc,3)])\n",
+ "\n",
+ "#Results\n",
+ "\n",
+ "print 'B(-3,4,0) in cartesian coordinates is',Bcart\n",
+ "print 'B(5,pi/2,-2) in cylindrical coordinates is',Bcyl"
+ ],
+ "language": "python",
+ "metadata": {},
+ "outputs": [
+ {
+ "output_type": "stream",
+ "stream": "stdout",
+ "text": [
+ "B(-3,4,0) in cartesian coordinates is [-2. 1. 0.]\n",
+ "B(5,pi/2,-2) in cylindrical coordinates is [ 2.467 1. 1.167]\n"
+ ]
+ }
+ ],
+ "prompt_number": 2
+ },
+ {
+ "cell_type": "markdown",
+ "metadata": {},
+ "source": [
+ "<h3>Example 2.3, Page number: 44<h3>"
+ ]
+ },
+ {
+ "cell_type": "code",
+ "collapsed": false,
+ "input": [
+ " \n",
+ "import scipy\n",
+ "from numpy import *\n",
+ "\n",
+ "#Variable Declaration\n",
+ "\n",
+ "E=array([-5,10,3]) #in cylindrical coordinates\n",
+ "F=array([1,2,-6]) #in cylindrical coordinates\n",
+ "P=array([5,scipy.pi/2,3]) #in cylindrical coordinates\n",
+ "\n",
+ "#Calculations\n",
+ "\n",
+ "exf=cross(E,F)\n",
+ "ansa=scipy.sqrt(dot(exf,exf)) #|EXF|\n",
+ "ay=array([round(scipy.sin(scipy.pi/2),0),\n",
+ " round(scipy.cos(scipy.pi/2),0),0])\n",
+ "ansb=dot(E,ay)*ay\n",
+ "modE=scipy.sqrt(dot(E,E))\n",
+ "az=array([0,0,1])\n",
+ "thetaEz=(180/scipy.pi)*arccos(dot(E,az)/modE) #in degrees\n",
+ "ansc=90-thetaEz #in degrees\n",
+ "\n",
+ "#Results\n",
+ "\n",
+ "print '|EXF| =',round(ansa,2)\n",
+ "print 'The vector component of E at P parallel to the line x=2,z=3 =',ansb,','\n",
+ "print 'in spherical coordinates'\n",
+ "print 'The angle E makes with the surface z = 3 at P =',round(ansc,2),'degrees'"
+ ],
+ "language": "python",
+ "metadata": {},
+ "outputs": [
+ {
+ "output_type": "stream",
+ "stream": "stdout",
+ "text": [
+ "|EXF| = 74.06\n",
+ "The vector component of E at P parallel to the line x=2,z=3 = [-5. -0. -0.] ,\n",
+ "in spherical coordinates\n",
+ "The angle E makes with the surface z = 3 at P = 15.02 degrees\n"
+ ]
+ }
+ ],
+ "prompt_number": 3
+ },
+ {
+ "cell_type": "markdown",
+ "metadata": {},
+ "source": [
+ "<h3>Example 2.4, Page number: 45<h3>"
+ ]
+ },
+ {
+ "cell_type": "code",
+ "collapsed": false,
+ "input": [
+ " \n",
+ "import scipy\n",
+ "from numpy import *\n",
+ "\n",
+ "#Variable Declaration\n",
+ "\n",
+ "aR=array([1,0,0]) #Unit vector along radial direction\n",
+ "ath=array([0,1,0]) #Unit vector along theta direction\n",
+ "aph=array([0,0,1]) #Unit vector along phi direction\n",
+ "P=array([10,scipy.pi*150/180,scipy.pi*330/180])\n",
+ "\n",
+ "#Calculations\n",
+ "\n",
+ "r=dot(P,aR)\n",
+ "q=dot(P,aph)\n",
+ "p=dot(P,ath)\n",
+ "R=r*scipy.sin(q)\n",
+ "Ph=-scipy.sin(p)*scipy.cos(q)/r\n",
+ "Q=r*r\n",
+ "D=array([R,Ph,Q]) #D at P(10,150\u00b0,330\u00b0)\n",
+ "rDr=round(dot(aR,D),0) #radial component of D\n",
+ "rDth=round(dot(-ath,D),3) #theta component of D\n",
+ "rDph=round(dot(aph,D),0) #phi component of D\n",
+ "\n",
+ "Dn=array([r*scipy.sin(q),0,0]) #Component of D normal to surface r=10\n",
+ "Dt=D-Dn #Component of D tangential to surface r=10\n",
+ "Dtr=round(dot(aR,Dt),0) #radial component of Dt\n",
+ "Dtth=round(dot(-ath,Dt),3) #theta component of Dt\n",
+ "Dtph=round(dot(aph,Dt),0) #phi component of Dt\n",
+ "rDt=array([Dtr,Dtth,Dtph])\n",
+ "\n",
+ " #Unit vector normal to D and tangential to cone theta=45 degrees\n",
+ "\n",
+ "U=cross(D,ath)\n",
+ "u=U/scipy.sqrt(dot(U,U)) \n",
+ "ru=array([round(dot(aR,u),4),round(dot(ath,u),4),round(dot(aph,u),4)])\n",
+ "\n",
+ "#Results\n",
+ "\n",
+ "print 'D at P(10,150\u00b0,330\u00b0) = [',rDr,' ',rDth,' ',rDph,']'\n",
+ "print 'The component of D tangential to the spherical surface r = 10 at P ='\n",
+ "print '[',Dtr,' ',Dtth,' ',Dtph,']'\n",
+ "print 'A unit vector at P perpendicular to D and tangential to cone 0 = 150\u00b0 ='\n",
+ "print ru"
+ ],
+ "language": "python",
+ "metadata": {},
+ "outputs": [
+ {
+ "output_type": "stream",
+ "stream": "stdout",
+ "text": [
+ "D at P(10,150\u00b0,330\u00b0) = [ -5.0 0.043 100.0 ]\n",
+ "The component of D tangential to the spherical surface r = 10 at P =\n",
+ "[ 0.0 0.043 100.0 ]\n",
+ "A unit vector at P perpendicular to D and tangential to cone 0 = 150\u00b0 =\n",
+ "[-0.9988 0. -0.0499]\n"
+ ]
+ }
+ ],
+ "prompt_number": 10
+ }
+ ],
+ "metadata": {}
+ }
+ ]
} \ No newline at end of file
generated by cgit (git 2.34.1) at 2025-04-07 09:00:28 +0000