{
"metadata": {
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"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": {}
}
]
}