mirror of
https://github.com/CoolProp/CoolProp.git
synced 2026-04-23 03:00:17 -04:00
1020 lines
69 KiB
Plaintext
1020 lines
69 KiB
Plaintext
{
|
|
"metadata": {
|
|
"name": "",
|
|
"signature": "sha256:ec1f215eb4df0e5e2bdc002acd107005a2401db562ea545eaf34be0f8fe96c93"
|
|
},
|
|
"nbformat": 3,
|
|
"nbformat_minor": 0,
|
|
"worksheets": [
|
|
{
|
|
"cells": [
|
|
{
|
|
"cell_type": "code",
|
|
"collapsed": false,
|
|
"input": [
|
|
"from __future__ import division\n",
|
|
"from sympy import *\n",
|
|
"from IPython.display import display, Math, Latex\n",
|
|
"from IPython.core.display import display_html\n",
|
|
"init_session(quiet=True, use_latex='mathjax')\n",
|
|
"init_printing()"
|
|
],
|
|
"language": "python",
|
|
"metadata": {},
|
|
"outputs": [
|
|
{
|
|
"output_type": "stream",
|
|
"stream": "stdout",
|
|
"text": [
|
|
"IPython console for SymPy 0.7.6 (Python 2.7.8-32-bit) (ground types: python)\n"
|
|
]
|
|
}
|
|
],
|
|
"prompt_number": 1
|
|
},
|
|
{
|
|
"cell_type": "code",
|
|
"collapsed": false,
|
|
"input": [
|
|
"def format_deriv(arg, itau, idel):\n",
|
|
" \"\"\" \n",
|
|
" A function for giving a nice latex representation of \n",
|
|
" the partial derivative in question \n",
|
|
" \"\"\"\n",
|
|
" if itau+idel == 1:\n",
|
|
" numexp = ''\n",
|
|
" else:\n",
|
|
" numexp = '^{{{s:d}}}'.format(s=itau+idel)\n",
|
|
" \n",
|
|
" if itau == 0:\n",
|
|
" tau = ''\n",
|
|
" elif itau == 1:\n",
|
|
" tau = '\\\\partial \\\\tau'\n",
|
|
" else:\n",
|
|
" tau = '\\\\partial \\\\tau^{{{s:d}}}'.format(s=itau)\n",
|
|
" \n",
|
|
" if idel == 0:\n",
|
|
" delta = ''\n",
|
|
" elif idel == 1:\n",
|
|
" delta = '\\\\partial \\\\delta'\n",
|
|
" else:\n",
|
|
" delta = '\\\\partial \\\\delta^{{{s:d}}}'.format(s=idel)\n",
|
|
" \n",
|
|
" temp = '\\\\frac{{\\\\partial{{{numexp:s}}} {arg:s}}}{{{{{tau:s}}}{{{delta:s}}}}} = '\n",
|
|
" return temp.format(**locals())"
|
|
],
|
|
"language": "python",
|
|
"metadata": {},
|
|
"outputs": [],
|
|
"prompt_number": 2
|
|
},
|
|
{
|
|
"cell_type": "heading",
|
|
"level": 1,
|
|
"metadata": {},
|
|
"source": [
|
|
"Derivatives of $\\Delta$"
|
|
]
|
|
},
|
|
{
|
|
"cell_type": "code",
|
|
"collapsed": false,
|
|
"input": [
|
|
"B_i, a_i, tau, delta, XX = symbols('B_i, a_i, tau, delta, XX')\n",
|
|
"theta = symbols('theta', cls=Function)(tau, delta)\n",
|
|
"Delta = theta**2+B_i*((delta-1)**2)**a_i\n",
|
|
"display(Math('\\\\Delta = ' + latex(Delta)))\n",
|
|
"\n",
|
|
"#c_i, beta_i = symbols('c_i, beta_i')\n",
|
|
"#_theta = (1-tau) + c_i*((delta-1)**2)**(1/(2*beta_i))\n",
|
|
"\n",
|
|
"def deriv(idel, itau):\n",
|
|
" dd = diff(diff(Delta, tau, itau), delta, idel)\n",
|
|
" dd = dd.subs(((delta-1)**2)**(a_i)/(delta-1)**idel, XX)\n",
|
|
" dd = factor(dd, XX)\n",
|
|
" dd = dd.subs(XX, ((delta-1)**2)**(a_i)/(delta-1)**idel)\n",
|
|
" \n",
|
|
" #dd = use(dd, simplify, 1)\n",
|
|
" display(Math(format_deriv('\\Delta',itau,idel)\n",
|
|
" + latex(dd)))\n",
|
|
" #el = diff(diff(Delta, tau, itau), delta, idel)\n",
|
|
" #args = dict(c_i = 0.7, beta_i = 0.3, B_i = 0.3, a_i = 3.5, tau = 1.3, delta = 0.9) \n",
|
|
" #for n in range(1,5):\n",
|
|
" # for _itau in range(0,n+1):\n",
|
|
" # s = diff(diff(_theta, tau, _itau), delta, n-_itau).subs(args)\n",
|
|
" # el = el.replace(diff(diff(theta, tau, _itau), delta, n-_itau), s)\n",
|
|
" #el = el.subs(theta, _theta)\n",
|
|
" #print el.subs(args).evalf()\n",
|
|
"\n",
|
|
"for n in range(1,5):\n",
|
|
" for itau in range(0,n+1):\n",
|
|
" deriv(itau, n-itau)"
|
|
],
|
|
"language": "python",
|
|
"metadata": {},
|
|
"outputs": [
|
|
{
|
|
"latex": [
|
|
"$$\\Delta = B_{i} \\left(\\left(\\delta - 1\\right)^{2}\\right)^{a_{i}} + \\theta^{2}{\\left (\\tau,\\delta \\right )}$$"
|
|
],
|
|
"metadata": {},
|
|
"output_type": "display_data",
|
|
"text": [
|
|
"<IPython.core.display.Math at 0xb95fcf8>"
|
|
]
|
|
},
|
|
{
|
|
"latex": [
|
|
"$$\\frac{\\partial{} \\Delta}{{\\partial \\tau}{}} = 2 \\theta{\\left (\\tau,\\delta \\right )} \\frac{\\partial}{\\partial \\tau} \\theta{\\left (\\tau,\\delta \\right )}$$"
|
|
],
|
|
"metadata": {},
|
|
"output_type": "display_data",
|
|
"text": [
|
|
"<IPython.core.display.Math at 0xb96fac8>"
|
|
]
|
|
},
|
|
{
|
|
"latex": [
|
|
"$$\\frac{\\partial{} \\Delta}{{}{\\partial \\delta}} = \\frac{2 B_{i} a_{i} \\left(\\left(\\delta - 1\\right)^{2}\\right)^{a_{i}}}{\\delta - 1} + 2 \\theta{\\left (\\tau,\\delta \\right )} \\frac{\\partial}{\\partial \\delta} \\theta{\\left (\\tau,\\delta \\right )}$$"
|
|
],
|
|
"metadata": {},
|
|
"output_type": "display_data",
|
|
"text": [
|
|
"<IPython.core.display.Math at 0xb929630>"
|
|
]
|
|
},
|
|
{
|
|
"latex": [
|
|
"$$\\frac{\\partial{^{2}} \\Delta}{{\\partial \\tau^{2}}{}} = 2 \\theta{\\left (\\tau,\\delta \\right )} \\frac{\\partial^{2}}{\\partial \\tau^{2}} \\theta{\\left (\\tau,\\delta \\right )} + 2 \\left(\\frac{\\partial}{\\partial \\tau} \\theta{\\left (\\tau,\\delta \\right )}\\right)^{2}$$"
|
|
],
|
|
"metadata": {},
|
|
"output_type": "display_data",
|
|
"text": [
|
|
"<IPython.core.display.Math at 0xbb55b00>"
|
|
]
|
|
},
|
|
{
|
|
"latex": [
|
|
"$$\\frac{\\partial{^{2}} \\Delta}{{\\partial \\tau}{\\partial \\delta}} = 2 \\left(\\theta{\\left (\\tau,\\delta \\right )} \\frac{\\partial^{2}}{\\partial \\delta\\partial \\tau} \\theta{\\left (\\tau,\\delta \\right )} + \\frac{\\partial}{\\partial \\delta} \\theta{\\left (\\tau,\\delta \\right )} \\frac{\\partial}{\\partial \\tau} \\theta{\\left (\\tau,\\delta \\right )}\\right)$$"
|
|
],
|
|
"metadata": {},
|
|
"output_type": "display_data",
|
|
"text": [
|
|
"<IPython.core.display.Math at 0xb6d1908>"
|
|
]
|
|
},
|
|
{
|
|
"latex": [
|
|
"$$\\frac{\\partial{^{2}} \\Delta}{{}{\\partial \\delta^{2}}} = 2 \\theta{\\left (\\tau,\\delta \\right )} \\frac{\\partial^{2}}{\\partial \\delta^{2}} \\theta{\\left (\\tau,\\delta \\right )} + 2 \\left(\\frac{\\partial}{\\partial \\delta} \\theta{\\left (\\tau,\\delta \\right )}\\right)^{2} + \\frac{2 \\left(\\left(\\delta - 1\\right)^{2}\\right)^{a_{i}}}{\\left(\\delta - 1\\right)^{2}} \\left(2 B_{i} a_{i}^{2} - B_{i} a_{i}\\right)$$"
|
|
],
|
|
"metadata": {},
|
|
"output_type": "display_data",
|
|
"text": [
|
|
"<IPython.core.display.Math at 0xcd1b6d8>"
|
|
]
|
|
},
|
|
{
|
|
"latex": [
|
|
"$$\\frac{\\partial{^{3}} \\Delta}{{\\partial \\tau^{3}}{}} = 2 \\theta{\\left (\\tau,\\delta \\right )} \\frac{\\partial^{3}}{\\partial \\tau^{3}} \\theta{\\left (\\tau,\\delta \\right )} + 6 \\frac{\\partial}{\\partial \\tau} \\theta{\\left (\\tau,\\delta \\right )} \\frac{\\partial^{2}}{\\partial \\tau^{2}} \\theta{\\left (\\tau,\\delta \\right )}$$"
|
|
],
|
|
"metadata": {},
|
|
"output_type": "display_data",
|
|
"text": [
|
|
"<IPython.core.display.Math at 0xcd1b898>"
|
|
]
|
|
},
|
|
{
|
|
"latex": [
|
|
"$$\\frac{\\partial{^{3}} \\Delta}{{\\partial \\tau^{2}}{\\partial \\delta}} = 2 \\left(\\theta{\\left (\\tau,\\delta \\right )} \\frac{\\partial^{3}}{\\partial \\delta\\partial \\tau^{2}} \\theta{\\left (\\tau,\\delta \\right )} + \\frac{\\partial}{\\partial \\delta} \\theta{\\left (\\tau,\\delta \\right )} \\frac{\\partial^{2}}{\\partial \\tau^{2}} \\theta{\\left (\\tau,\\delta \\right )} + 2 \\frac{\\partial}{\\partial \\tau} \\theta{\\left (\\tau,\\delta \\right )} \\frac{\\partial^{2}}{\\partial \\delta\\partial \\tau} \\theta{\\left (\\tau,\\delta \\right )}\\right)$$"
|
|
],
|
|
"metadata": {},
|
|
"output_type": "display_data",
|
|
"text": [
|
|
"<IPython.core.display.Math at 0xcd1b3c8>"
|
|
]
|
|
},
|
|
{
|
|
"latex": [
|
|
"$$\\frac{\\partial{^{3}} \\Delta}{{\\partial \\tau}{\\partial \\delta^{2}}} = 2 \\theta{\\left (\\tau,\\delta \\right )} \\frac{\\partial^{3}}{\\partial \\delta^{2}\\partial \\tau} \\theta{\\left (\\tau,\\delta \\right )} + 4 \\frac{\\partial}{\\partial \\delta} \\theta{\\left (\\tau,\\delta \\right )} \\frac{\\partial^{2}}{\\partial \\delta\\partial \\tau} \\theta{\\left (\\tau,\\delta \\right )} + 2 \\frac{\\partial}{\\partial \\tau} \\theta{\\left (\\tau,\\delta \\right )} \\frac{\\partial^{2}}{\\partial \\delta^{2}} \\theta{\\left (\\tau,\\delta \\right )}$$"
|
|
],
|
|
"metadata": {},
|
|
"output_type": "display_data",
|
|
"text": [
|
|
"<IPython.core.display.Math at 0xb929630>"
|
|
]
|
|
},
|
|
{
|
|
"latex": [
|
|
"$$\\frac{\\partial{^{3}} \\Delta}{{}{\\partial \\delta^{3}}} = 2 \\theta{\\left (\\tau,\\delta \\right )} \\frac{\\partial^{3}}{\\partial \\delta^{3}} \\theta{\\left (\\tau,\\delta \\right )} + 6 \\frac{\\partial}{\\partial \\delta} \\theta{\\left (\\tau,\\delta \\right )} \\frac{\\partial^{2}}{\\partial \\delta^{2}} \\theta{\\left (\\tau,\\delta \\right )} + \\frac{2 \\left(\\left(\\delta - 1\\right)^{2}\\right)^{a_{i}}}{\\left(\\delta - 1\\right)^{3}} \\left(4 B_{i} a_{i}^{3} - 6 B_{i} a_{i}^{2} + 2 B_{i} a_{i}\\right)$$"
|
|
],
|
|
"metadata": {},
|
|
"output_type": "display_data",
|
|
"text": [
|
|
"<IPython.core.display.Math at 0xbabd7f0>"
|
|
]
|
|
},
|
|
{
|
|
"latex": [
|
|
"$$\\frac{\\partial{^{4}} \\Delta}{{\\partial \\tau^{4}}{}} = 2 \\theta{\\left (\\tau,\\delta \\right )} \\frac{\\partial^{4}}{\\partial \\tau^{4}} \\theta{\\left (\\tau,\\delta \\right )} + 8 \\frac{\\partial}{\\partial \\tau} \\theta{\\left (\\tau,\\delta \\right )} \\frac{\\partial^{3}}{\\partial \\tau^{3}} \\theta{\\left (\\tau,\\delta \\right )} + 6 \\left(\\frac{\\partial^{2}}{\\partial \\tau^{2}} \\theta{\\left (\\tau,\\delta \\right )}\\right)^{2}$$"
|
|
],
|
|
"metadata": {},
|
|
"output_type": "display_data",
|
|
"text": [
|
|
"<IPython.core.display.Math at 0xba8def0>"
|
|
]
|
|
},
|
|
{
|
|
"latex": [
|
|
"$$\\frac{\\partial{^{4}} \\Delta}{{\\partial \\tau^{3}}{\\partial \\delta}} = 2 \\left(\\theta{\\left (\\tau,\\delta \\right )} \\frac{\\partial^{4}}{\\partial \\delta\\partial \\tau^{3}} \\theta{\\left (\\tau,\\delta \\right )} + \\frac{\\partial}{\\partial \\delta} \\theta{\\left (\\tau,\\delta \\right )} \\frac{\\partial^{3}}{\\partial \\tau^{3}} \\theta{\\left (\\tau,\\delta \\right )} + 3 \\frac{\\partial}{\\partial \\tau} \\theta{\\left (\\tau,\\delta \\right )} \\frac{\\partial^{3}}{\\partial \\delta\\partial \\tau^{2}} \\theta{\\left (\\tau,\\delta \\right )} + 3 \\frac{\\partial^{2}}{\\partial \\delta\\partial \\tau} \\theta{\\left (\\tau,\\delta \\right )} \\frac{\\partial^{2}}{\\partial \\tau^{2}} \\theta{\\left (\\tau,\\delta \\right )}\\right)$$"
|
|
],
|
|
"metadata": {},
|
|
"output_type": "display_data",
|
|
"text": [
|
|
"<IPython.core.display.Math at 0xba8d550>"
|
|
]
|
|
},
|
|
{
|
|
"latex": [
|
|
"$$\\frac{\\partial{^{4}} \\Delta}{{\\partial \\tau^{2}}{\\partial \\delta^{2}}} = 2 \\theta{\\left (\\tau,\\delta \\right )} \\frac{\\partial^{4}}{\\partial \\delta^{2}\\partial \\tau^{2}} \\theta{\\left (\\tau,\\delta \\right )} + 4 \\frac{\\partial}{\\partial \\delta} \\theta{\\left (\\tau,\\delta \\right )} \\frac{\\partial^{3}}{\\partial \\delta\\partial \\tau^{2}} \\theta{\\left (\\tau,\\delta \\right )} + 4 \\frac{\\partial}{\\partial \\tau} \\theta{\\left (\\tau,\\delta \\right )} \\frac{\\partial^{3}}{\\partial \\delta^{2}\\partial \\tau} \\theta{\\left (\\tau,\\delta \\right )} + 2 \\frac{\\partial^{2}}{\\partial \\delta^{2}} \\theta{\\left (\\tau,\\delta \\right )} \\frac{\\partial^{2}}{\\partial \\tau^{2}} \\theta{\\left (\\tau,\\delta \\right )} + 4 \\left(\\frac{\\partial^{2}}{\\partial \\delta\\partial \\tau} \\theta{\\left (\\tau,\\delta \\right )}\\right)^{2}$$"
|
|
],
|
|
"metadata": {},
|
|
"output_type": "display_data",
|
|
"text": [
|
|
"<IPython.core.display.Math at 0xbb55cc0>"
|
|
]
|
|
},
|
|
{
|
|
"latex": [
|
|
"$$\\frac{\\partial{^{4}} \\Delta}{{\\partial \\tau}{\\partial \\delta^{3}}} = 2 \\theta{\\left (\\tau,\\delta \\right )} \\frac{\\partial^{4}}{\\partial \\delta^{3}\\partial \\tau} \\theta{\\left (\\tau,\\delta \\right )} + 6 \\frac{\\partial}{\\partial \\delta} \\theta{\\left (\\tau,\\delta \\right )} \\frac{\\partial^{3}}{\\partial \\delta^{2}\\partial \\tau} \\theta{\\left (\\tau,\\delta \\right )} + 2 \\frac{\\partial}{\\partial \\tau} \\theta{\\left (\\tau,\\delta \\right )} \\frac{\\partial^{3}}{\\partial \\delta^{3}} \\theta{\\left (\\tau,\\delta \\right )} + 6 \\frac{\\partial^{2}}{\\partial \\delta^{2}} \\theta{\\left (\\tau,\\delta \\right )} \\frac{\\partial^{2}}{\\partial \\delta\\partial \\tau} \\theta{\\left (\\tau,\\delta \\right )}$$"
|
|
],
|
|
"metadata": {},
|
|
"output_type": "display_data",
|
|
"text": [
|
|
"<IPython.core.display.Math at 0xb9d6d30>"
|
|
]
|
|
},
|
|
{
|
|
"latex": [
|
|
"$$\\frac{\\partial{^{4}} \\Delta}{{}{\\partial \\delta^{4}}} = 2 \\theta{\\left (\\tau,\\delta \\right )} \\frac{\\partial^{4}}{\\partial \\delta^{4}} \\theta{\\left (\\tau,\\delta \\right )} + 8 \\frac{\\partial}{\\partial \\delta} \\theta{\\left (\\tau,\\delta \\right )} \\frac{\\partial^{3}}{\\partial \\delta^{3}} \\theta{\\left (\\tau,\\delta \\right )} + 6 \\left(\\frac{\\partial^{2}}{\\partial \\delta^{2}} \\theta{\\left (\\tau,\\delta \\right )}\\right)^{2} + \\frac{2 \\left(\\left(\\delta - 1\\right)^{2}\\right)^{a_{i}}}{\\left(\\delta - 1\\right)^{4}} \\left(8 B_{i} a_{i}^{4} - 24 B_{i} a_{i}^{3} + 22 B_{i} a_{i}^{2} - 6 B_{i} a_{i}\\right)$$"
|
|
],
|
|
"metadata": {},
|
|
"output_type": "display_data",
|
|
"text": [
|
|
"<IPython.core.display.Math at 0xb95f908>"
|
|
]
|
|
}
|
|
],
|
|
"prompt_number": 30
|
|
},
|
|
{
|
|
"cell_type": "heading",
|
|
"level": 1,
|
|
"metadata": {},
|
|
"source": [
|
|
"Derivatives of $\\theta$"
|
|
]
|
|
},
|
|
{
|
|
"cell_type": "code",
|
|
"collapsed": false,
|
|
"input": [
|
|
"A_i, beta_i, tau, delta = symbols('A_i, beta_i, tau, delta')\n",
|
|
"theta = (1-tau) + A_i*((delta-1)**2)**(1/(2*beta_i))\n",
|
|
"display(Math('\\\\theta = ' + latex(theta)))\n",
|
|
"\n",
|
|
"def deriv(idel, itau):\n",
|
|
" display(Math(format_deriv('\\\\theta',itau,idel)\n",
|
|
" + latex( diff(diff(theta, tau, itau), delta, idel)) ))\n",
|
|
" #print diff(diff(theta, tau, itau), delta, idel).subs(dict(A_i = 0.7, beta_i = 0.3, tau = 1.3, delta = 0.9))\n",
|
|
"\n",
|
|
"for n in range(1,5):\n",
|
|
" for itau in range(0,n+1):\n",
|
|
" deriv(itau, n-itau)"
|
|
],
|
|
"language": "python",
|
|
"metadata": {},
|
|
"outputs": [
|
|
{
|
|
"latex": [
|
|
"$$\\theta = A_{i} \\left(\\left(\\delta - 1\\right)^{2}\\right)^{\\frac{1}{2 \\beta_{i}}} - \\tau + 1$$"
|
|
],
|
|
"metadata": {},
|
|
"output_type": "display_data",
|
|
"text": [
|
|
"<IPython.core.display.Math at 0xb713358>"
|
|
]
|
|
},
|
|
{
|
|
"latex": [
|
|
"$$\\frac{\\partial{} \\theta}{{\\partial \\tau}{}} = -1$$"
|
|
],
|
|
"metadata": {},
|
|
"output_type": "display_data",
|
|
"text": [
|
|
"<IPython.core.display.Math at 0xb713710>"
|
|
]
|
|
},
|
|
{
|
|
"latex": [
|
|
"$$\\frac{\\partial{} \\theta}{{}{\\partial \\delta}} = \\frac{A_{i} \\left(2 \\delta - 2\\right) \\left(\\left(\\delta - 1\\right)^{2}\\right)^{\\frac{1}{2 \\beta_{i}}}}{2 \\beta_{i} \\left(\\delta - 1\\right)^{2}}$$"
|
|
],
|
|
"metadata": {},
|
|
"output_type": "display_data",
|
|
"text": [
|
|
"<IPython.core.display.Math at 0xb80eb70>"
|
|
]
|
|
},
|
|
{
|
|
"latex": [
|
|
"$$\\frac{\\partial{^{2}} \\theta}{{\\partial \\tau^{2}}{}} = 0$$"
|
|
],
|
|
"metadata": {},
|
|
"output_type": "display_data",
|
|
"text": [
|
|
"<IPython.core.display.Math at 0xb713898>"
|
|
]
|
|
},
|
|
{
|
|
"latex": [
|
|
"$$\\frac{\\partial{^{2}} \\theta}{{\\partial \\tau}{\\partial \\delta}} = 0$$"
|
|
],
|
|
"metadata": {},
|
|
"output_type": "display_data",
|
|
"text": [
|
|
"<IPython.core.display.Math at 0xb713f98>"
|
|
]
|
|
},
|
|
{
|
|
"latex": [
|
|
"$$\\frac{\\partial{^{2}} \\theta}{{}{\\partial \\delta^{2}}} = \\frac{A_{i} \\left(-1 + \\frac{1}{\\beta_{i}}\\right) \\left(\\left(\\delta - 1\\right)^{2}\\right)^{\\frac{1}{2 \\beta_{i}}}}{\\beta_{i} \\left(\\delta - 1\\right)^{2}}$$"
|
|
],
|
|
"metadata": {},
|
|
"output_type": "display_data",
|
|
"text": [
|
|
"<IPython.core.display.Math at 0xb718390>"
|
|
]
|
|
},
|
|
{
|
|
"latex": [
|
|
"$$\\frac{\\partial{^{3}} \\theta}{{\\partial \\tau^{3}}{}} = 0$$"
|
|
],
|
|
"metadata": {},
|
|
"output_type": "display_data",
|
|
"text": [
|
|
"<IPython.core.display.Math at 0xb8a7390>"
|
|
]
|
|
},
|
|
{
|
|
"latex": [
|
|
"$$\\frac{\\partial{^{3}} \\theta}{{\\partial \\tau^{2}}{\\partial \\delta}} = 0$$"
|
|
],
|
|
"metadata": {},
|
|
"output_type": "display_data",
|
|
"text": [
|
|
"<IPython.core.display.Math at 0xb8a7b38>"
|
|
]
|
|
},
|
|
{
|
|
"latex": [
|
|
"$$\\frac{\\partial{^{3}} \\theta}{{\\partial \\tau}{\\partial \\delta^{2}}} = 0$$"
|
|
],
|
|
"metadata": {},
|
|
"output_type": "display_data",
|
|
"text": [
|
|
"<IPython.core.display.Math at 0xb8a79b0>"
|
|
]
|
|
},
|
|
{
|
|
"latex": [
|
|
"$$\\frac{\\partial{^{3}} \\theta}{{}{\\partial \\delta^{3}}} = \\frac{A_{i} \\left(\\left(\\delta - 1\\right)^{2}\\right)^{\\frac{1}{2 \\beta_{i}}}}{\\beta_{i} \\left(\\delta - 1\\right)^{3}} \\left(2 - \\frac{3}{\\beta_{i}} + \\frac{1}{\\beta_{i}^{2}}\\right)$$"
|
|
],
|
|
"metadata": {},
|
|
"output_type": "display_data",
|
|
"text": [
|
|
"<IPython.core.display.Math at 0xb8a7f60>"
|
|
]
|
|
},
|
|
{
|
|
"latex": [
|
|
"$$\\frac{\\partial{^{4}} \\theta}{{\\partial \\tau^{4}}{}} = 0$$"
|
|
],
|
|
"metadata": {},
|
|
"output_type": "display_data",
|
|
"text": [
|
|
"<IPython.core.display.Math at 0xb8abd68>"
|
|
]
|
|
},
|
|
{
|
|
"latex": [
|
|
"$$\\frac{\\partial{^{4}} \\theta}{{\\partial \\tau^{3}}{\\partial \\delta}} = 0$$"
|
|
],
|
|
"metadata": {},
|
|
"output_type": "display_data",
|
|
"text": [
|
|
"<IPython.core.display.Math at 0xb8b0978>"
|
|
]
|
|
},
|
|
{
|
|
"latex": [
|
|
"$$\\frac{\\partial{^{4}} \\theta}{{\\partial \\tau^{2}}{\\partial \\delta^{2}}} = 0$$"
|
|
],
|
|
"metadata": {},
|
|
"output_type": "display_data",
|
|
"text": [
|
|
"<IPython.core.display.Math at 0xb8b06d8>"
|
|
]
|
|
},
|
|
{
|
|
"latex": [
|
|
"$$\\frac{\\partial{^{4}} \\theta}{{\\partial \\tau}{\\partial \\delta^{3}}} = 0$$"
|
|
],
|
|
"metadata": {},
|
|
"output_type": "display_data",
|
|
"text": [
|
|
"<IPython.core.display.Math at 0xb8b04e0>"
|
|
]
|
|
},
|
|
{
|
|
"latex": [
|
|
"$$\\frac{\\partial{^{4}} \\theta}{{}{\\partial \\delta^{4}}} = \\frac{A_{i} \\left(\\left(\\delta - 1\\right)^{2}\\right)^{\\frac{1}{2 \\beta_{i}}}}{\\beta_{i} \\left(\\delta - 1\\right)^{4}} \\left(-6 + \\frac{11}{\\beta_{i}} - \\frac{6}{\\beta_{i}^{2}} + \\frac{1}{\\beta_{i}^{3}}\\right)$$"
|
|
],
|
|
"metadata": {},
|
|
"output_type": "display_data",
|
|
"text": [
|
|
"<IPython.core.display.Math at 0xb8b9b38>"
|
|
]
|
|
}
|
|
],
|
|
"prompt_number": 6
|
|
},
|
|
{
|
|
"cell_type": "heading",
|
|
"level": 1,
|
|
"metadata": {},
|
|
"source": [
|
|
"Derivatives of $\\psi$"
|
|
]
|
|
},
|
|
{
|
|
"cell_type": "code",
|
|
"collapsed": false,
|
|
"input": [
|
|
"C_i, D_i, tau, delta = symbols('C_i, D_i, tau, delta')\n",
|
|
"psi = exp( -C_i*(delta-1)**2 -D_i*(tau-1)**2)\n",
|
|
"display(Math('\\\\psi = ' + latex(psi)))\n",
|
|
"\n",
|
|
"def deriv(idel, itau):\n",
|
|
" display(Math(format_deriv('\\\\psi',itau, idel)\n",
|
|
" + latex( diff(diff(psi, tau, itau), delta, idel)).replace(latex(psi),'\\psi') ))\n",
|
|
" #print diff(diff(psi, tau, itau), delta, idel).subs(dict(C_i = 10, D_i = 275, tau= 1.3, delta = 0.9))\n",
|
|
"\n",
|
|
"for n in range(1,5):\n",
|
|
" for itau in range(0,n+1):\n",
|
|
" deriv(itau, n-itau)"
|
|
],
|
|
"language": "python",
|
|
"metadata": {},
|
|
"outputs": [
|
|
{
|
|
"latex": [
|
|
"$$\\psi = e^{- C_{i} \\left(\\delta - 1\\right)^{2} - D_{i} \\left(\\tau - 1\\right)^{2}}$$"
|
|
],
|
|
"metadata": {},
|
|
"output_type": "display_data",
|
|
"text": [
|
|
"<IPython.core.display.Math at 0xb8b0a90>"
|
|
]
|
|
},
|
|
{
|
|
"latex": [
|
|
"$$\\frac{\\partial{} \\psi}{{\\partial \\tau}{}} = - D_{i} \\left(2 \\tau - 2\\right) \\psi$$"
|
|
],
|
|
"metadata": {},
|
|
"output_type": "display_data",
|
|
"text": [
|
|
"<IPython.core.display.Math at 0xb8b06d8>"
|
|
]
|
|
},
|
|
{
|
|
"latex": [
|
|
"$$\\frac{\\partial{} \\psi}{{}{\\partial \\delta}} = - C_{i} \\left(2 \\delta - 2\\right) \\psi$$"
|
|
],
|
|
"metadata": {},
|
|
"output_type": "display_data",
|
|
"text": [
|
|
"<IPython.core.display.Math at 0xb8db1d0>"
|
|
]
|
|
},
|
|
{
|
|
"latex": [
|
|
"$$\\frac{\\partial{^{2}} \\psi}{{\\partial \\tau^{2}}{}} = 2 D_{i} \\left(2 D_{i} \\left(\\tau - 1\\right)^{2} - 1\\right) \\psi$$"
|
|
],
|
|
"metadata": {},
|
|
"output_type": "display_data",
|
|
"text": [
|
|
"<IPython.core.display.Math at 0xb8d9d30>"
|
|
]
|
|
},
|
|
{
|
|
"latex": [
|
|
"$$\\frac{\\partial{^{2}} \\psi}{{\\partial \\tau}{\\partial \\delta}} = C_{i} D_{i} \\left(2 \\delta - 2\\right) \\left(2 \\tau - 2\\right) \\psi$$"
|
|
],
|
|
"metadata": {},
|
|
"output_type": "display_data",
|
|
"text": [
|
|
"<IPython.core.display.Math at 0xb8cfa90>"
|
|
]
|
|
},
|
|
{
|
|
"latex": [
|
|
"$$\\frac{\\partial{^{2}} \\psi}{{}{\\partial \\delta^{2}}} = 2 C_{i} \\left(2 C_{i} \\left(\\delta - 1\\right)^{2} - 1\\right) \\psi$$"
|
|
],
|
|
"metadata": {},
|
|
"output_type": "display_data",
|
|
"text": [
|
|
"<IPython.core.display.Math at 0xb8dbfd0>"
|
|
]
|
|
},
|
|
{
|
|
"latex": [
|
|
"$$\\frac{\\partial{^{3}} \\psi}{{\\partial \\tau^{3}}{}} = 4 D_{i}^{2} \\left(\\tau - 1\\right) \\left(- 2 D_{i} \\left(\\tau - 1\\right)^{2} + 3\\right) \\psi$$"
|
|
],
|
|
"metadata": {},
|
|
"output_type": "display_data",
|
|
"text": [
|
|
"<IPython.core.display.Math at 0xb8d16d8>"
|
|
]
|
|
},
|
|
{
|
|
"latex": [
|
|
"$$\\frac{\\partial{^{3}} \\psi}{{\\partial \\tau^{2}}{\\partial \\delta}} = - 2 C_{i} D_{i} \\left(2 \\delta - 2\\right) \\left(2 D_{i} \\left(\\tau - 1\\right)^{2} - 1\\right) \\psi$$"
|
|
],
|
|
"metadata": {},
|
|
"output_type": "display_data",
|
|
"text": [
|
|
"<IPython.core.display.Math at 0xb8d1b38>"
|
|
]
|
|
},
|
|
{
|
|
"latex": [
|
|
"$$\\frac{\\partial{^{3}} \\psi}{{\\partial \\tau}{\\partial \\delta^{2}}} = 4 C_{i} D_{i} \\left(\\tau - 1\\right) \\left(- 2 C_{i} \\left(\\delta - 1\\right)^{2} + 1\\right) \\psi$$"
|
|
],
|
|
"metadata": {},
|
|
"output_type": "display_data",
|
|
"text": [
|
|
"<IPython.core.display.Math at 0xb8d9d30>"
|
|
]
|
|
},
|
|
{
|
|
"latex": [
|
|
"$$\\frac{\\partial{^{3}} \\psi}{{}{\\partial \\delta^{3}}} = 4 C_{i}^{2} \\left(\\delta - 1\\right) \\left(- 2 C_{i} \\left(\\delta - 1\\right)^{2} + 3\\right) \\psi$$"
|
|
],
|
|
"metadata": {},
|
|
"output_type": "display_data",
|
|
"text": [
|
|
"<IPython.core.display.Math at 0xb8dbc88>"
|
|
]
|
|
},
|
|
{
|
|
"latex": [
|
|
"$$\\frac{\\partial{^{4}} \\psi}{{\\partial \\tau^{4}}{}} = 4 D_{i}^{2} \\left(4 D_{i}^{2} \\left(\\tau - 1\\right)^{4} - 12 D_{i} \\left(\\tau - 1\\right)^{2} + 3\\right) \\psi$$"
|
|
],
|
|
"metadata": {},
|
|
"output_type": "display_data",
|
|
"text": [
|
|
"<IPython.core.display.Math at 0xb8b06d8>"
|
|
]
|
|
},
|
|
{
|
|
"latex": [
|
|
"$$\\frac{\\partial{^{4}} \\psi}{{\\partial \\tau^{3}}{\\partial \\delta}} = - 4 C_{i} D_{i}^{2} \\left(2 \\delta - 2\\right) \\left(\\tau - 1\\right) \\left(- 2 D_{i} \\left(\\tau - 1\\right)^{2} + 3\\right) \\psi$$"
|
|
],
|
|
"metadata": {},
|
|
"output_type": "display_data",
|
|
"text": [
|
|
"<IPython.core.display.Math at 0xb8b04e0>"
|
|
]
|
|
},
|
|
{
|
|
"latex": [
|
|
"$$\\frac{\\partial{^{4}} \\psi}{{\\partial \\tau^{2}}{\\partial \\delta^{2}}} = 4 C_{i} D_{i} \\left(2 C_{i} \\left(\\delta - 1\\right)^{2} - 1\\right) \\left(2 D_{i} \\left(\\tau - 1\\right)^{2} - 1\\right) \\psi$$"
|
|
],
|
|
"metadata": {},
|
|
"output_type": "display_data",
|
|
"text": [
|
|
"<IPython.core.display.Math at 0x41e00b8>"
|
|
]
|
|
},
|
|
{
|
|
"latex": [
|
|
"$$\\frac{\\partial{^{4}} \\psi}{{\\partial \\tau}{\\partial \\delta^{3}}} = 8 C_{i}^{2} D_{i} \\left(\\delta - 1\\right) \\left(\\tau - 1\\right) \\left(2 C_{i} \\left(\\delta - 1\\right)^{2} - 3\\right) \\psi$$"
|
|
],
|
|
"metadata": {},
|
|
"output_type": "display_data",
|
|
"text": [
|
|
"<IPython.core.display.Math at 0xb86cb70>"
|
|
]
|
|
},
|
|
{
|
|
"latex": [
|
|
"$$\\frac{\\partial{^{4}} \\psi}{{}{\\partial \\delta^{4}}} = 4 C_{i}^{2} \\left(4 C_{i}^{2} \\left(\\delta - 1\\right)^{4} - 12 C_{i} \\left(\\delta - 1\\right)^{2} + 3\\right) \\psi$$"
|
|
],
|
|
"metadata": {},
|
|
"output_type": "display_data",
|
|
"text": [
|
|
"<IPython.core.display.Math at 0xb8b06d8>"
|
|
]
|
|
}
|
|
],
|
|
"prompt_number": 8
|
|
},
|
|
{
|
|
"cell_type": "heading",
|
|
"level": 1,
|
|
"metadata": {},
|
|
"source": [
|
|
"Derivatives of $\\alpha_r$"
|
|
]
|
|
},
|
|
{
|
|
"cell_type": "code",
|
|
"collapsed": false,
|
|
"input": [
|
|
"n_i, tau, delta = symbols('n_i, tau, delta')\n",
|
|
"Delta_bi = symbols('Delta_bi', cls=Function)(tau, delta)\n",
|
|
"psi = symbols('psi', cls=Function)(tau, delta)\n",
|
|
"alphar = n_i*delta*Delta_bi*psi\n",
|
|
"display(Math('\\\\alpha^{{r}}_{{NA,i}} = ' \n",
|
|
" + latex(alphar).replace(latex(Delta_bi),'\\Delta^{b_i}')))\n",
|
|
"\n",
|
|
"def collector(a):\n",
|
|
" return collect(a, Delta_bi)\n",
|
|
" \n",
|
|
"def deriv(idel, itau):\n",
|
|
" dd = diff(diff(alphar, tau, itau), delta, idel)\n",
|
|
" dd = simplify(dd)\n",
|
|
" dd = use(dd, expand, 2)\n",
|
|
" display(Math(format_deriv('\\\\alpha^{{r}}_{{NA,i}}',itau,idel)\n",
|
|
" + latex(dd).replace(latex(Delta_bi),'\\Delta^{b_i}') ))\n",
|
|
"\n",
|
|
"for n in range(1,5):\n",
|
|
" for itau in range(0,n+1):\n",
|
|
" deriv(itau, n-itau)"
|
|
],
|
|
"language": "python",
|
|
"metadata": {},
|
|
"outputs": [
|
|
{
|
|
"latex": [
|
|
"$$\\alpha^{{r}}_{{NA,i}} = \\delta n_{i} \\Delta^{b_i} \\psi{\\left (\\tau,\\delta \\right )}$$"
|
|
],
|
|
"metadata": {},
|
|
"output_type": "display_data",
|
|
"text": [
|
|
"<IPython.core.display.Math at 0xb61dd68>"
|
|
]
|
|
},
|
|
{
|
|
"latex": [
|
|
"$$\\frac{\\partial{} \\alpha^{{r}}_{{NA,i}}}{{\\partial \\tau}{}} = \\delta n_{i} \\left(\\Delta^{b_i} \\frac{\\partial}{\\partial \\tau} \\psi{\\left (\\tau,\\delta \\right )} + \\psi{\\left (\\tau,\\delta \\right )} \\frac{\\partial}{\\partial \\tau} \\Delta^{b_i}\\right)$$"
|
|
],
|
|
"metadata": {},
|
|
"output_type": "display_data",
|
|
"text": [
|
|
"<IPython.core.display.Math at 0xb86cef0>"
|
|
]
|
|
},
|
|
{
|
|
"latex": [
|
|
"$$\\frac{\\partial{} \\alpha^{{r}}_{{NA,i}}}{{}{\\partial \\delta}} = n_{i} \\left(\\delta \\Delta^{b_i} \\frac{\\partial}{\\partial \\delta} \\psi{\\left (\\tau,\\delta \\right )} + \\delta \\psi{\\left (\\tau,\\delta \\right )} \\frac{\\partial}{\\partial \\delta} \\Delta^{b_i} + \\Delta^{b_i} \\psi{\\left (\\tau,\\delta \\right )}\\right)$$"
|
|
],
|
|
"metadata": {},
|
|
"output_type": "display_data",
|
|
"text": [
|
|
"<IPython.core.display.Math at 0xb9e4f28>"
|
|
]
|
|
},
|
|
{
|
|
"latex": [
|
|
"$$\\frac{\\partial{^{2}} \\alpha^{{r}}_{{NA,i}}}{{\\partial \\tau^{2}}{}} = \\delta n_{i} \\left(\\Delta^{b_i} \\frac{\\partial^{2}}{\\partial \\tau^{2}} \\psi{\\left (\\tau,\\delta \\right )} + \\psi{\\left (\\tau,\\delta \\right )} \\frac{\\partial^{2}}{\\partial \\tau^{2}} \\Delta^{b_i} + 2 \\frac{\\partial}{\\partial \\tau} \\Delta^{b_i} \\frac{\\partial}{\\partial \\tau} \\psi{\\left (\\tau,\\delta \\right )}\\right)$$"
|
|
],
|
|
"metadata": {},
|
|
"output_type": "display_data",
|
|
"text": [
|
|
"<IPython.core.display.Math at 0xb9d9940>"
|
|
]
|
|
},
|
|
{
|
|
"latex": [
|
|
"$$\\frac{\\partial{^{2}} \\alpha^{{r}}_{{NA,i}}}{{\\partial \\tau}{\\partial \\delta}} = n_{i} \\left(\\delta \\Delta^{b_i} \\frac{\\partial^{2}}{\\partial \\delta\\partial \\tau} \\psi{\\left (\\tau,\\delta \\right )} + \\delta \\psi{\\left (\\tau,\\delta \\right )} \\frac{\\partial^{2}}{\\partial \\delta\\partial \\tau} \\Delta^{b_i} + \\delta \\frac{\\partial}{\\partial \\delta} \\Delta^{b_i} \\frac{\\partial}{\\partial \\tau} \\psi{\\left (\\tau,\\delta \\right )} + \\delta \\frac{\\partial}{\\partial \\tau} \\Delta^{b_i} \\frac{\\partial}{\\partial \\delta} \\psi{\\left (\\tau,\\delta \\right )} + \\Delta^{b_i} \\frac{\\partial}{\\partial \\tau} \\psi{\\left (\\tau,\\delta \\right )} + \\psi{\\left (\\tau,\\delta \\right )} \\frac{\\partial}{\\partial \\tau} \\Delta^{b_i}\\right)$$"
|
|
],
|
|
"metadata": {},
|
|
"output_type": "display_data",
|
|
"text": [
|
|
"<IPython.core.display.Math at 0xb9530b8>"
|
|
]
|
|
},
|
|
{
|
|
"latex": [
|
|
"$$\\frac{\\partial{^{2}} \\alpha^{{r}}_{{NA,i}}}{{}{\\partial \\delta^{2}}} = n_{i} \\left(\\delta \\Delta^{b_i} \\frac{\\partial^{2}}{\\partial \\delta^{2}} \\psi{\\left (\\tau,\\delta \\right )} + \\delta \\psi{\\left (\\tau,\\delta \\right )} \\frac{\\partial^{2}}{\\partial \\delta^{2}} \\Delta^{b_i} + 2 \\delta \\frac{\\partial}{\\partial \\delta} \\Delta^{b_i} \\frac{\\partial}{\\partial \\delta} \\psi{\\left (\\tau,\\delta \\right )} + 2 \\Delta^{b_i} \\frac{\\partial}{\\partial \\delta} \\psi{\\left (\\tau,\\delta \\right )} + 2 \\psi{\\left (\\tau,\\delta \\right )} \\frac{\\partial}{\\partial \\delta} \\Delta^{b_i}\\right)$$"
|
|
],
|
|
"metadata": {},
|
|
"output_type": "display_data",
|
|
"text": [
|
|
"<IPython.core.display.Math at 0xb9e4cf8>"
|
|
]
|
|
},
|
|
{
|
|
"latex": [
|
|
"$$\\frac{\\partial{^{3}} \\alpha^{{r}}_{{NA,i}}}{{\\partial \\tau^{3}}{}} = \\delta n_{i} \\left(\\Delta^{b_i} \\frac{\\partial^{3}}{\\partial \\tau^{3}} \\psi{\\left (\\tau,\\delta \\right )} + \\psi{\\left (\\tau,\\delta \\right )} \\frac{\\partial^{3}}{\\partial \\tau^{3}} \\Delta^{b_i} + 3 \\frac{\\partial}{\\partial \\tau} \\Delta^{b_i} \\frac{\\partial^{2}}{\\partial \\tau^{2}} \\psi{\\left (\\tau,\\delta \\right )} + 3 \\frac{\\partial}{\\partial \\tau} \\psi{\\left (\\tau,\\delta \\right )} \\frac{\\partial^{2}}{\\partial \\tau^{2}} \\Delta^{b_i}\\right)$$"
|
|
],
|
|
"metadata": {},
|
|
"output_type": "display_data",
|
|
"text": [
|
|
"<IPython.core.display.Math at 0xb9d9940>"
|
|
]
|
|
},
|
|
{
|
|
"latex": [
|
|
"$$\\frac{\\partial{^{3}} \\alpha^{{r}}_{{NA,i}}}{{\\partial \\tau^{2}}{\\partial \\delta}} = n_{i} \\left(\\delta \\Delta^{b_i} \\frac{\\partial^{3}}{\\partial \\delta\\partial \\tau^{2}} \\psi{\\left (\\tau,\\delta \\right )} + \\delta \\psi{\\left (\\tau,\\delta \\right )} \\frac{\\partial^{3}}{\\partial \\delta\\partial \\tau^{2}} \\Delta^{b_i} + \\delta \\frac{\\partial}{\\partial \\delta} \\Delta^{b_i} \\frac{\\partial^{2}}{\\partial \\tau^{2}} \\psi{\\left (\\tau,\\delta \\right )} + 2 \\delta \\frac{\\partial}{\\partial \\tau} \\Delta^{b_i} \\frac{\\partial^{2}}{\\partial \\delta\\partial \\tau} \\psi{\\left (\\tau,\\delta \\right )} + \\delta \\frac{\\partial}{\\partial \\delta} \\psi{\\left (\\tau,\\delta \\right )} \\frac{\\partial^{2}}{\\partial \\tau^{2}} \\Delta^{b_i} + 2 \\delta \\frac{\\partial}{\\partial \\tau} \\psi{\\left (\\tau,\\delta \\right )} \\frac{\\partial^{2}}{\\partial \\delta\\partial \\tau} \\Delta^{b_i} + \\Delta^{b_i} \\frac{\\partial^{2}}{\\partial \\tau^{2}} \\psi{\\left (\\tau,\\delta \\right )} + \\psi{\\left (\\tau,\\delta \\right )} \\frac{\\partial^{2}}{\\partial \\tau^{2}} \\Delta^{b_i} + 2 \\frac{\\partial}{\\partial \\tau} \\Delta^{b_i} \\frac{\\partial}{\\partial \\tau} \\psi{\\left (\\tau,\\delta \\right )}\\right)$$"
|
|
],
|
|
"metadata": {},
|
|
"output_type": "display_data",
|
|
"text": [
|
|
"<IPython.core.display.Math at 0xb93d630>"
|
|
]
|
|
},
|
|
{
|
|
"latex": [
|
|
"$$\\frac{\\partial{^{3}} \\alpha^{{r}}_{{NA,i}}}{{\\partial \\tau}{\\partial \\delta^{2}}} = n_{i} \\left(\\delta \\Delta^{b_i} \\frac{\\partial^{3}}{\\partial \\delta^{2}\\partial \\tau} \\psi{\\left (\\tau,\\delta \\right )} + \\delta \\psi{\\left (\\tau,\\delta \\right )} \\frac{\\partial^{3}}{\\partial \\delta^{2}\\partial \\tau} \\Delta^{b_i} + 2 \\delta \\frac{\\partial}{\\partial \\delta} \\Delta^{b_i} \\frac{\\partial^{2}}{\\partial \\delta\\partial \\tau} \\psi{\\left (\\tau,\\delta \\right )} + \\delta \\frac{\\partial}{\\partial \\tau} \\Delta^{b_i} \\frac{\\partial^{2}}{\\partial \\delta^{2}} \\psi{\\left (\\tau,\\delta \\right )} + 2 \\delta \\frac{\\partial}{\\partial \\delta} \\psi{\\left (\\tau,\\delta \\right )} \\frac{\\partial^{2}}{\\partial \\delta\\partial \\tau} \\Delta^{b_i} + \\delta \\frac{\\partial}{\\partial \\tau} \\psi{\\left (\\tau,\\delta \\right )} \\frac{\\partial^{2}}{\\partial \\delta^{2}} \\Delta^{b_i} + 2 \\Delta^{b_i} \\frac{\\partial^{2}}{\\partial \\delta\\partial \\tau} \\psi{\\left (\\tau,\\delta \\right )} + 2 \\psi{\\left (\\tau,\\delta \\right )} \\frac{\\partial^{2}}{\\partial \\delta\\partial \\tau} \\Delta^{b_i} + 2 \\frac{\\partial}{\\partial \\delta} \\Delta^{b_i} \\frac{\\partial}{\\partial \\tau} \\psi{\\left (\\tau,\\delta \\right )} + 2 \\frac{\\partial}{\\partial \\tau} \\Delta^{b_i} \\frac{\\partial}{\\partial \\delta} \\psi{\\left (\\tau,\\delta \\right )}\\right)$$"
|
|
],
|
|
"metadata": {},
|
|
"output_type": "display_data",
|
|
"text": [
|
|
"<IPython.core.display.Math at 0x23b2198>"
|
|
]
|
|
},
|
|
{
|
|
"latex": [
|
|
"$$\\frac{\\partial{^{3}} \\alpha^{{r}}_{{NA,i}}}{{}{\\partial \\delta^{3}}} = n_{i} \\left(\\delta \\Delta^{b_i} \\frac{\\partial^{3}}{\\partial \\delta^{3}} \\psi{\\left (\\tau,\\delta \\right )} + \\delta \\psi{\\left (\\tau,\\delta \\right )} \\frac{\\partial^{3}}{\\partial \\delta^{3}} \\Delta^{b_i} + 3 \\delta \\frac{\\partial}{\\partial \\delta} \\Delta^{b_i} \\frac{\\partial^{2}}{\\partial \\delta^{2}} \\psi{\\left (\\tau,\\delta \\right )} + 3 \\delta \\frac{\\partial}{\\partial \\delta} \\psi{\\left (\\tau,\\delta \\right )} \\frac{\\partial^{2}}{\\partial \\delta^{2}} \\Delta^{b_i} + 3 \\Delta^{b_i} \\frac{\\partial^{2}}{\\partial \\delta^{2}} \\psi{\\left (\\tau,\\delta \\right )} + 3 \\psi{\\left (\\tau,\\delta \\right )} \\frac{\\partial^{2}}{\\partial \\delta^{2}} \\Delta^{b_i} + 6 \\frac{\\partial}{\\partial \\delta} \\Delta^{b_i} \\frac{\\partial}{\\partial \\delta} \\psi{\\left (\\tau,\\delta \\right )}\\right)$$"
|
|
],
|
|
"metadata": {},
|
|
"output_type": "display_data",
|
|
"text": [
|
|
"<IPython.core.display.Math at 0xb9462b0>"
|
|
]
|
|
},
|
|
{
|
|
"latex": [
|
|
"$$\\frac{\\partial{^{4}} \\alpha^{{r}}_{{NA,i}}}{{\\partial \\tau^{4}}{}} = \\delta n_{i} \\left(\\Delta^{b_i} \\frac{\\partial^{4}}{\\partial \\tau^{4}} \\psi{\\left (\\tau,\\delta \\right )} + \\psi{\\left (\\tau,\\delta \\right )} \\frac{\\partial^{4}}{\\partial \\tau^{4}} \\Delta^{b_i} + 4 \\frac{\\partial}{\\partial \\tau} \\Delta^{b_i} \\frac{\\partial^{3}}{\\partial \\tau^{3}} \\psi{\\left (\\tau,\\delta \\right )} + 4 \\frac{\\partial}{\\partial \\tau} \\psi{\\left (\\tau,\\delta \\right )} \\frac{\\partial^{3}}{\\partial \\tau^{3}} \\Delta^{b_i} + 6 \\frac{\\partial^{2}}{\\partial \\tau^{2}} \\Delta^{b_i} \\frac{\\partial^{2}}{\\partial \\tau^{2}} \\psi{\\left (\\tau,\\delta \\right )}\\right)$$"
|
|
],
|
|
"metadata": {},
|
|
"output_type": "display_data",
|
|
"text": [
|
|
"<IPython.core.display.Math at 0xb86c240>"
|
|
]
|
|
},
|
|
{
|
|
"latex": [
|
|
"$$\\frac{\\partial{^{4}} \\alpha^{{r}}_{{NA,i}}}{{\\partial \\tau^{3}}{\\partial \\delta}} = n_{i} \\left(\\delta \\Delta^{b_i} \\frac{\\partial^{4}}{\\partial \\delta\\partial \\tau^{3}} \\psi{\\left (\\tau,\\delta \\right )} + \\delta \\psi{\\left (\\tau,\\delta \\right )} \\frac{\\partial^{4}}{\\partial \\delta\\partial \\tau^{3}} \\Delta^{b_i} + \\delta \\frac{\\partial}{\\partial \\delta} \\Delta^{b_i} \\frac{\\partial^{3}}{\\partial \\tau^{3}} \\psi{\\left (\\tau,\\delta \\right )} + 3 \\delta \\frac{\\partial}{\\partial \\tau} \\Delta^{b_i} \\frac{\\partial^{3}}{\\partial \\delta\\partial \\tau^{2}} \\psi{\\left (\\tau,\\delta \\right )} + \\delta \\frac{\\partial}{\\partial \\delta} \\psi{\\left (\\tau,\\delta \\right )} \\frac{\\partial^{3}}{\\partial \\tau^{3}} \\Delta^{b_i} + 3 \\delta \\frac{\\partial}{\\partial \\tau} \\psi{\\left (\\tau,\\delta \\right )} \\frac{\\partial^{3}}{\\partial \\delta\\partial \\tau^{2}} \\Delta^{b_i} + 3 \\delta \\frac{\\partial^{2}}{\\partial \\delta\\partial \\tau} \\Delta^{b_i} \\frac{\\partial^{2}}{\\partial \\tau^{2}} \\psi{\\left (\\tau,\\delta \\right )} + 3 \\delta \\frac{\\partial^{2}}{\\partial \\tau^{2}} \\Delta^{b_i} \\frac{\\partial^{2}}{\\partial \\delta\\partial \\tau} \\psi{\\left (\\tau,\\delta \\right )} + \\Delta^{b_i} \\frac{\\partial^{3}}{\\partial \\tau^{3}} \\psi{\\left (\\tau,\\delta \\right )} + \\psi{\\left (\\tau,\\delta \\right )} \\frac{\\partial^{3}}{\\partial \\tau^{3}} \\Delta^{b_i} + 3 \\frac{\\partial}{\\partial \\tau} \\Delta^{b_i} \\frac{\\partial^{2}}{\\partial \\tau^{2}} \\psi{\\left (\\tau,\\delta \\right )} + 3 \\frac{\\partial}{\\partial \\tau} \\psi{\\left (\\tau,\\delta \\right )} \\frac{\\partial^{2}}{\\partial \\tau^{2}} \\Delta^{b_i}\\right)$$"
|
|
],
|
|
"metadata": {},
|
|
"output_type": "display_data",
|
|
"text": [
|
|
"<IPython.core.display.Math at 0xb9e4ef0>"
|
|
]
|
|
},
|
|
{
|
|
"latex": [
|
|
"$$\\frac{\\partial{^{4}} \\alpha^{{r}}_{{NA,i}}}{{\\partial \\tau^{2}}{\\partial \\delta^{2}}} = n_{i} \\left(\\delta \\Delta^{b_i} \\frac{\\partial^{4}}{\\partial \\delta^{2}\\partial \\tau^{2}} \\psi{\\left (\\tau,\\delta \\right )} + \\delta \\psi{\\left (\\tau,\\delta \\right )} \\frac{\\partial^{4}}{\\partial \\delta^{2}\\partial \\tau^{2}} \\Delta^{b_i} + 2 \\delta \\frac{\\partial}{\\partial \\delta} \\Delta^{b_i} \\frac{\\partial^{3}}{\\partial \\delta\\partial \\tau^{2}} \\psi{\\left (\\tau,\\delta \\right )} + 2 \\delta \\frac{\\partial}{\\partial \\tau} \\Delta^{b_i} \\frac{\\partial^{3}}{\\partial \\delta^{2}\\partial \\tau} \\psi{\\left (\\tau,\\delta \\right )} + 2 \\delta \\frac{\\partial}{\\partial \\delta} \\psi{\\left (\\tau,\\delta \\right )} \\frac{\\partial^{3}}{\\partial \\delta\\partial \\tau^{2}} \\Delta^{b_i} + 2 \\delta \\frac{\\partial}{\\partial \\tau} \\psi{\\left (\\tau,\\delta \\right )} \\frac{\\partial^{3}}{\\partial \\delta^{2}\\partial \\tau} \\Delta^{b_i} + \\delta \\frac{\\partial^{2}}{\\partial \\delta^{2}} \\Delta^{b_i} \\frac{\\partial^{2}}{\\partial \\tau^{2}} \\psi{\\left (\\tau,\\delta \\right )} + 4 \\delta \\frac{\\partial^{2}}{\\partial \\delta\\partial \\tau} \\Delta^{b_i} \\frac{\\partial^{2}}{\\partial \\delta\\partial \\tau} \\psi{\\left (\\tau,\\delta \\right )} + \\delta \\frac{\\partial^{2}}{\\partial \\tau^{2}} \\Delta^{b_i} \\frac{\\partial^{2}}{\\partial \\delta^{2}} \\psi{\\left (\\tau,\\delta \\right )} + 2 \\Delta^{b_i} \\frac{\\partial^{3}}{\\partial \\delta\\partial \\tau^{2}} \\psi{\\left (\\tau,\\delta \\right )} + 2 \\psi{\\left (\\tau,\\delta \\right )} \\frac{\\partial^{3}}{\\partial \\delta\\partial \\tau^{2}} \\Delta^{b_i} + 2 \\frac{\\partial}{\\partial \\delta} \\Delta^{b_i} \\frac{\\partial^{2}}{\\partial \\tau^{2}} \\psi{\\left (\\tau,\\delta \\right )} + 4 \\frac{\\partial}{\\partial \\tau} \\Delta^{b_i} \\frac{\\partial^{2}}{\\partial \\delta\\partial \\tau} \\psi{\\left (\\tau,\\delta \\right )} + 2 \\frac{\\partial}{\\partial \\delta} \\psi{\\left (\\tau,\\delta \\right )} \\frac{\\partial^{2}}{\\partial \\tau^{2}} \\Delta^{b_i} + 4 \\frac{\\partial}{\\partial \\tau} \\psi{\\left (\\tau,\\delta \\right )} \\frac{\\partial^{2}}{\\partial \\delta\\partial \\tau} \\Delta^{b_i}\\right)$$"
|
|
],
|
|
"metadata": {},
|
|
"output_type": "display_data",
|
|
"text": [
|
|
"<IPython.core.display.Math at 0xb9e4d68>"
|
|
]
|
|
},
|
|
{
|
|
"latex": [
|
|
"$$\\frac{\\partial{^{4}} \\alpha^{{r}}_{{NA,i}}}{{\\partial \\tau}{\\partial \\delta^{3}}} = n_{i} \\left(\\delta \\Delta^{b_i} \\frac{\\partial^{4}}{\\partial \\delta^{3}\\partial \\tau} \\psi{\\left (\\tau,\\delta \\right )} + \\delta \\psi{\\left (\\tau,\\delta \\right )} \\frac{\\partial^{4}}{\\partial \\delta^{3}\\partial \\tau} \\Delta^{b_i} + 3 \\delta \\frac{\\partial}{\\partial \\delta} \\Delta^{b_i} \\frac{\\partial^{3}}{\\partial \\delta^{2}\\partial \\tau} \\psi{\\left (\\tau,\\delta \\right )} + \\delta \\frac{\\partial}{\\partial \\tau} \\Delta^{b_i} \\frac{\\partial^{3}}{\\partial \\delta^{3}} \\psi{\\left (\\tau,\\delta \\right )} + 3 \\delta \\frac{\\partial}{\\partial \\delta} \\psi{\\left (\\tau,\\delta \\right )} \\frac{\\partial^{3}}{\\partial \\delta^{2}\\partial \\tau} \\Delta^{b_i} + \\delta \\frac{\\partial}{\\partial \\tau} \\psi{\\left (\\tau,\\delta \\right )} \\frac{\\partial^{3}}{\\partial \\delta^{3}} \\Delta^{b_i} + 3 \\delta \\frac{\\partial^{2}}{\\partial \\delta^{2}} \\Delta^{b_i} \\frac{\\partial^{2}}{\\partial \\delta\\partial \\tau} \\psi{\\left (\\tau,\\delta \\right )} + 3 \\delta \\frac{\\partial^{2}}{\\partial \\delta\\partial \\tau} \\Delta^{b_i} \\frac{\\partial^{2}}{\\partial \\delta^{2}} \\psi{\\left (\\tau,\\delta \\right )} + 3 \\Delta^{b_i} \\frac{\\partial^{3}}{\\partial \\delta^{2}\\partial \\tau} \\psi{\\left (\\tau,\\delta \\right )} + 3 \\psi{\\left (\\tau,\\delta \\right )} \\frac{\\partial^{3}}{\\partial \\delta^{2}\\partial \\tau} \\Delta^{b_i} + 6 \\frac{\\partial}{\\partial \\delta} \\Delta^{b_i} \\frac{\\partial^{2}}{\\partial \\delta\\partial \\tau} \\psi{\\left (\\tau,\\delta \\right )} + 3 \\frac{\\partial}{\\partial \\tau} \\Delta^{b_i} \\frac{\\partial^{2}}{\\partial \\delta^{2}} \\psi{\\left (\\tau,\\delta \\right )} + 6 \\frac{\\partial}{\\partial \\delta} \\psi{\\left (\\tau,\\delta \\right )} \\frac{\\partial^{2}}{\\partial \\delta\\partial \\tau} \\Delta^{b_i} + 3 \\frac{\\partial}{\\partial \\tau} \\psi{\\left (\\tau,\\delta \\right )} \\frac{\\partial^{2}}{\\partial \\delta^{2}} \\Delta^{b_i}\\right)$$"
|
|
],
|
|
"metadata": {},
|
|
"output_type": "display_data",
|
|
"text": [
|
|
"<IPython.core.display.Math at 0xb946080>"
|
|
]
|
|
},
|
|
{
|
|
"latex": [
|
|
"$$\\frac{\\partial{^{4}} \\alpha^{{r}}_{{NA,i}}}{{}{\\partial \\delta^{4}}} = n_{i} \\left(\\delta \\Delta^{b_i} \\frac{\\partial^{4}}{\\partial \\delta^{4}} \\psi{\\left (\\tau,\\delta \\right )} + \\delta \\psi{\\left (\\tau,\\delta \\right )} \\frac{\\partial^{4}}{\\partial \\delta^{4}} \\Delta^{b_i} + 4 \\delta \\frac{\\partial}{\\partial \\delta} \\Delta^{b_i} \\frac{\\partial^{3}}{\\partial \\delta^{3}} \\psi{\\left (\\tau,\\delta \\right )} + 4 \\delta \\frac{\\partial}{\\partial \\delta} \\psi{\\left (\\tau,\\delta \\right )} \\frac{\\partial^{3}}{\\partial \\delta^{3}} \\Delta^{b_i} + 6 \\delta \\frac{\\partial^{2}}{\\partial \\delta^{2}} \\Delta^{b_i} \\frac{\\partial^{2}}{\\partial \\delta^{2}} \\psi{\\left (\\tau,\\delta \\right )} + 4 \\Delta^{b_i} \\frac{\\partial^{3}}{\\partial \\delta^{3}} \\psi{\\left (\\tau,\\delta \\right )} + 4 \\psi{\\left (\\tau,\\delta \\right )} \\frac{\\partial^{3}}{\\partial \\delta^{3}} \\Delta^{b_i} + 12 \\frac{\\partial}{\\partial \\delta} \\Delta^{b_i} \\frac{\\partial^{2}}{\\partial \\delta^{2}} \\psi{\\left (\\tau,\\delta \\right )} + 12 \\frac{\\partial}{\\partial \\delta} \\psi{\\left (\\tau,\\delta \\right )} \\frac{\\partial^{2}}{\\partial \\delta^{2}} \\Delta^{b_i}\\right)$$"
|
|
],
|
|
"metadata": {},
|
|
"output_type": "display_data",
|
|
"text": [
|
|
"<IPython.core.display.Math at 0xb86c6a0>"
|
|
]
|
|
}
|
|
],
|
|
"prompt_number": 9
|
|
},
|
|
{
|
|
"cell_type": "heading",
|
|
"level": 1,
|
|
"metadata": {},
|
|
"source": [
|
|
"Derivatives of $\\Delta^{b_i}$"
|
|
]
|
|
},
|
|
{
|
|
"cell_type": "code",
|
|
"collapsed": false,
|
|
"input": [
|
|
"b_i, tau, delta = symbols('b_i, tau, delta')\n",
|
|
"Delta = symbols('Delta', cls=Function)(tau, delta)\n",
|
|
"\n",
|
|
"#c_i, beta_i, a_i, d_i = symbols('c_i, beta_i, a_i, d_i')\n",
|
|
"#_theta = (1-tau) + c_i*((delta-1)**2)**(1/(2*beta_i))\n",
|
|
"#_Delta = _theta**2+d_i*((delta-1)**2)**a_i\n",
|
|
"#print _Delta\n",
|
|
"\n",
|
|
"def deriv(idel, itau):\n",
|
|
" dd = diff(diff(Delta**b_i, tau, itau), delta, idel)\n",
|
|
" dd = use(dd, simplify, 2)\n",
|
|
" dd = use(dd, factor, 2)\n",
|
|
" \n",
|
|
" display(Math(format_deriv('\\\\Delta^{{b_i}}', itau, idel)\n",
|
|
" + latex(simplify(dd)) ))\n",
|
|
" \n",
|
|
" #el = diff(diff(Delta**b_i, tau, itau), delta, idel)\n",
|
|
" #args = dict(c_i = 0.7, beta_i = 0.3, d_i = 0.3, a_i = 3.5, b_i = 0.875, tau = 1.3, delta = 0.9) \n",
|
|
" #for n in range(1,5):\n",
|
|
" # for _itau in range(0,n+1):\n",
|
|
" # s = diff(diff(_Delta, tau, _itau), delta, n-_itau).subs(args)\n",
|
|
" # el = el.replace(diff(diff(Delta, tau, _itau), delta, n-_itau), s)\n",
|
|
" #el = el.subs(Delta, _Delta)\n",
|
|
" #print el.subs(args).evalf()\n",
|
|
"\n",
|
|
"for n in range(1,5):\n",
|
|
" for itau in range(0,n+1):\n",
|
|
" deriv(itau, n-itau)"
|
|
],
|
|
"language": "python",
|
|
"metadata": {},
|
|
"outputs": [
|
|
{
|
|
"latex": [
|
|
"$$\\frac{\\partial{} \\Delta^{{b_i}}}{{\\partial \\tau}{}} = b_{i} \\Delta^{b_{i} - 1}{\\left (\\tau,\\delta \\right )} \\frac{\\partial}{\\partial \\tau} \\Delta{\\left (\\tau,\\delta \\right )}$$"
|
|
],
|
|
"metadata": {},
|
|
"output_type": "display_data",
|
|
"text": [
|
|
"<IPython.core.display.Math at 0xb946e80>"
|
|
]
|
|
},
|
|
{
|
|
"latex": [
|
|
"$$\\frac{\\partial{} \\Delta^{{b_i}}}{{}{\\partial \\delta}} = b_{i} \\Delta^{b_{i} - 1}{\\left (\\tau,\\delta \\right )} \\frac{\\partial}{\\partial \\delta} \\Delta{\\left (\\tau,\\delta \\right )}$$"
|
|
],
|
|
"metadata": {},
|
|
"output_type": "display_data",
|
|
"text": [
|
|
"<IPython.core.display.Math at 0xb93da20>"
|
|
]
|
|
},
|
|
{
|
|
"latex": [
|
|
"$$\\frac{\\partial{^{2}} \\Delta^{{b_i}}}{{\\partial \\tau^{2}}{}} = b_{i} \\left(b_{i} \\left(\\frac{\\partial}{\\partial \\tau} \\Delta{\\left (\\tau,\\delta \\right )}\\right)^{2} + \\Delta{\\left (\\tau,\\delta \\right )} \\frac{\\partial^{2}}{\\partial \\tau^{2}} \\Delta{\\left (\\tau,\\delta \\right )} - \\left(\\frac{\\partial}{\\partial \\tau} \\Delta{\\left (\\tau,\\delta \\right )}\\right)^{2}\\right) \\Delta^{b_{i} - 2}{\\left (\\tau,\\delta \\right )}$$"
|
|
],
|
|
"metadata": {},
|
|
"output_type": "display_data",
|
|
"text": [
|
|
"<IPython.core.display.Math at 0xb953240>"
|
|
]
|
|
},
|
|
{
|
|
"latex": [
|
|
"$$\\frac{\\partial{^{2}} \\Delta^{{b_i}}}{{\\partial \\tau}{\\partial \\delta}} = \\frac{b_{i}}{\\Delta^{3}{\\left (\\tau,\\delta \\right )}} \\left(\\left(b_{i} - 1\\right) \\Delta^{b_{i} + 1}{\\left (\\tau,\\delta \\right )} \\frac{\\partial}{\\partial \\delta} \\Delta{\\left (\\tau,\\delta \\right )} \\frac{\\partial}{\\partial \\tau} \\Delta{\\left (\\tau,\\delta \\right )} + \\Delta^{b_{i} + 2}{\\left (\\tau,\\delta \\right )} \\frac{\\partial^{2}}{\\partial \\delta\\partial \\tau} \\Delta{\\left (\\tau,\\delta \\right )}\\right)$$"
|
|
],
|
|
"metadata": {},
|
|
"output_type": "display_data",
|
|
"text": [
|
|
"<IPython.core.display.Math at 0xb946ba8>"
|
|
]
|
|
},
|
|
{
|
|
"latex": [
|
|
"$$\\frac{\\partial{^{2}} \\Delta^{{b_i}}}{{}{\\partial \\delta^{2}}} = b_{i} \\left(b_{i} \\left(\\frac{\\partial}{\\partial \\delta} \\Delta{\\left (\\tau,\\delta \\right )}\\right)^{2} + \\Delta{\\left (\\tau,\\delta \\right )} \\frac{\\partial^{2}}{\\partial \\delta^{2}} \\Delta{\\left (\\tau,\\delta \\right )} - \\left(\\frac{\\partial}{\\partial \\delta} \\Delta{\\left (\\tau,\\delta \\right )}\\right)^{2}\\right) \\Delta^{b_{i} - 2}{\\left (\\tau,\\delta \\right )}$$"
|
|
],
|
|
"metadata": {},
|
|
"output_type": "display_data",
|
|
"text": [
|
|
"<IPython.core.display.Math at 0xc4ddc50>"
|
|
]
|
|
},
|
|
{
|
|
"latex": [
|
|
"$$\\frac{\\partial{^{3}} \\Delta^{{b_i}}}{{\\partial \\tau^{3}}{}} = b_{i} \\left(3 \\left(b_{i} - 1\\right) \\Delta{\\left (\\tau,\\delta \\right )} \\frac{\\partial}{\\partial \\tau} \\Delta{\\left (\\tau,\\delta \\right )} \\frac{\\partial^{2}}{\\partial \\tau^{2}} \\Delta{\\left (\\tau,\\delta \\right )} + \\left(b_{i}^{2} - 3 b_{i} + 2\\right) \\left(\\frac{\\partial}{\\partial \\tau} \\Delta{\\left (\\tau,\\delta \\right )}\\right)^{3} + \\Delta^{2}{\\left (\\tau,\\delta \\right )} \\frac{\\partial^{3}}{\\partial \\tau^{3}} \\Delta{\\left (\\tau,\\delta \\right )}\\right) \\Delta^{b_{i} - 3}{\\left (\\tau,\\delta \\right )}$$"
|
|
],
|
|
"metadata": {},
|
|
"output_type": "display_data",
|
|
"text": [
|
|
"<IPython.core.display.Math at 0xb9f97f0>"
|
|
]
|
|
},
|
|
{
|
|
"latex": [
|
|
"$$\\frac{\\partial{^{3}} \\Delta^{{b_i}}}{{\\partial \\tau^{2}}{\\partial \\delta}} = b_{i} \\left(b_{i}^{2} \\frac{\\partial}{\\partial \\delta} \\Delta{\\left (\\tau,\\delta \\right )} \\left(\\frac{\\partial}{\\partial \\tau} \\Delta{\\left (\\tau,\\delta \\right )}\\right)^{2} + b_{i} \\Delta{\\left (\\tau,\\delta \\right )} \\frac{\\partial}{\\partial \\delta} \\Delta{\\left (\\tau,\\delta \\right )} \\frac{\\partial^{2}}{\\partial \\tau^{2}} \\Delta{\\left (\\tau,\\delta \\right )} + 2 b_{i} \\Delta{\\left (\\tau,\\delta \\right )} \\frac{\\partial}{\\partial \\tau} \\Delta{\\left (\\tau,\\delta \\right )} \\frac{\\partial^{2}}{\\partial \\delta\\partial \\tau} \\Delta{\\left (\\tau,\\delta \\right )} - 3 b_{i} \\frac{\\partial}{\\partial \\delta} \\Delta{\\left (\\tau,\\delta \\right )} \\left(\\frac{\\partial}{\\partial \\tau} \\Delta{\\left (\\tau,\\delta \\right )}\\right)^{2} + \\Delta^{2}{\\left (\\tau,\\delta \\right )} \\frac{\\partial^{3}}{\\partial \\delta\\partial \\tau^{2}} \\Delta{\\left (\\tau,\\delta \\right )} - \\Delta{\\left (\\tau,\\delta \\right )} \\frac{\\partial}{\\partial \\delta} \\Delta{\\left (\\tau,\\delta \\right )} \\frac{\\partial^{2}}{\\partial \\tau^{2}} \\Delta{\\left (\\tau,\\delta \\right )} - 2 \\Delta{\\left (\\tau,\\delta \\right )} \\frac{\\partial}{\\partial \\tau} \\Delta{\\left (\\tau,\\delta \\right )} \\frac{\\partial^{2}}{\\partial \\delta\\partial \\tau} \\Delta{\\left (\\tau,\\delta \\right )} + 2 \\frac{\\partial}{\\partial \\delta} \\Delta{\\left (\\tau,\\delta \\right )} \\left(\\frac{\\partial}{\\partial \\tau} \\Delta{\\left (\\tau,\\delta \\right )}\\right)^{2}\\right) \\Delta^{b_{i} - 3}{\\left (\\tau,\\delta \\right )}$$"
|
|
],
|
|
"metadata": {},
|
|
"output_type": "display_data",
|
|
"text": [
|
|
"<IPython.core.display.Math at 0xc4ddba8>"
|
|
]
|
|
},
|
|
{
|
|
"latex": [
|
|
"$$\\frac{\\partial{^{3}} \\Delta^{{b_i}}}{{\\partial \\tau}{\\partial \\delta^{2}}} = b_{i} \\left(\\left(b_{i}^{2} - 3 b_{i} + 2\\right) \\left(\\frac{\\partial}{\\partial \\delta} \\Delta{\\left (\\tau,\\delta \\right )}\\right)^{2} \\frac{\\partial}{\\partial \\tau} \\Delta{\\left (\\tau,\\delta \\right )} + \\left(2 b_{i} \\frac{\\partial}{\\partial \\delta} \\Delta{\\left (\\tau,\\delta \\right )} \\frac{\\partial^{2}}{\\partial \\delta\\partial \\tau} \\Delta{\\left (\\tau,\\delta \\right )} + b_{i} \\frac{\\partial}{\\partial \\tau} \\Delta{\\left (\\tau,\\delta \\right )} \\frac{\\partial^{2}}{\\partial \\delta^{2}} \\Delta{\\left (\\tau,\\delta \\right )} - 2 \\frac{\\partial}{\\partial \\delta} \\Delta{\\left (\\tau,\\delta \\right )} \\frac{\\partial^{2}}{\\partial \\delta\\partial \\tau} \\Delta{\\left (\\tau,\\delta \\right )} - \\frac{\\partial}{\\partial \\tau} \\Delta{\\left (\\tau,\\delta \\right )} \\frac{\\partial^{2}}{\\partial \\delta^{2}} \\Delta{\\left (\\tau,\\delta \\right )}\\right) \\Delta{\\left (\\tau,\\delta \\right )} + \\Delta^{2}{\\left (\\tau,\\delta \\right )} \\frac{\\partial^{3}}{\\partial \\delta^{2}\\partial \\tau} \\Delta{\\left (\\tau,\\delta \\right )}\\right) \\Delta^{b_{i} - 3}{\\left (\\tau,\\delta \\right )}$$"
|
|
],
|
|
"metadata": {},
|
|
"output_type": "display_data",
|
|
"text": [
|
|
"<IPython.core.display.Math at 0xb9e4710>"
|
|
]
|
|
},
|
|
{
|
|
"latex": [
|
|
"$$\\frac{\\partial{^{3}} \\Delta^{{b_i}}}{{}{\\partial \\delta^{3}}} = b_{i} \\left(3 \\left(b_{i} - 1\\right) \\Delta{\\left (\\tau,\\delta \\right )} \\frac{\\partial}{\\partial \\delta} \\Delta{\\left (\\tau,\\delta \\right )} \\frac{\\partial^{2}}{\\partial \\delta^{2}} \\Delta{\\left (\\tau,\\delta \\right )} + \\left(b_{i}^{2} - 3 b_{i} + 2\\right) \\left(\\frac{\\partial}{\\partial \\delta} \\Delta{\\left (\\tau,\\delta \\right )}\\right)^{3} + \\Delta^{2}{\\left (\\tau,\\delta \\right )} \\frac{\\partial^{3}}{\\partial \\delta^{3}} \\Delta{\\left (\\tau,\\delta \\right )}\\right) \\Delta^{b_{i} - 3}{\\left (\\tau,\\delta \\right )}$$"
|
|
],
|
|
"metadata": {},
|
|
"output_type": "display_data",
|
|
"text": [
|
|
"<IPython.core.display.Math at 0xb9e4a58>"
|
|
]
|
|
},
|
|
{
|
|
"latex": [
|
|
"$$\\frac{\\partial{^{4}} \\Delta^{{b_i}}}{{\\partial \\tau^{4}}{}} = b_{i} \\left(6 \\left(b_{i}^{2} - 3 b_{i} + 2\\right) \\Delta{\\left (\\tau,\\delta \\right )} \\left(\\frac{\\partial}{\\partial \\tau} \\Delta{\\left (\\tau,\\delta \\right )}\\right)^{2} \\frac{\\partial^{2}}{\\partial \\tau^{2}} \\Delta{\\left (\\tau,\\delta \\right )} + \\left(b_{i}^{3} - 6 b_{i}^{2} + 11 b_{i} - 6\\right) \\left(\\frac{\\partial}{\\partial \\tau} \\Delta{\\left (\\tau,\\delta \\right )}\\right)^{4} + \\left(4 b_{i} \\frac{\\partial}{\\partial \\tau} \\Delta{\\left (\\tau,\\delta \\right )} \\frac{\\partial^{3}}{\\partial \\tau^{3}} \\Delta{\\left (\\tau,\\delta \\right )} + 3 b_{i} \\left(\\frac{\\partial^{2}}{\\partial \\tau^{2}} \\Delta{\\left (\\tau,\\delta \\right )}\\right)^{2} - 4 \\frac{\\partial}{\\partial \\tau} \\Delta{\\left (\\tau,\\delta \\right )} \\frac{\\partial^{3}}{\\partial \\tau^{3}} \\Delta{\\left (\\tau,\\delta \\right )} - 3 \\left(\\frac{\\partial^{2}}{\\partial \\tau^{2}} \\Delta{\\left (\\tau,\\delta \\right )}\\right)^{2}\\right) \\Delta^{2}{\\left (\\tau,\\delta \\right )} + \\Delta^{3}{\\left (\\tau,\\delta \\right )} \\frac{\\partial^{4}}{\\partial \\tau^{4}} \\Delta{\\left (\\tau,\\delta \\right )}\\right) \\Delta^{b_{i} - 4}{\\left (\\tau,\\delta \\right )}$$"
|
|
],
|
|
"metadata": {},
|
|
"output_type": "display_data",
|
|
"text": [
|
|
"<IPython.core.display.Math at 0xb953240>"
|
|
]
|
|
},
|
|
{
|
|
"latex": [
|
|
"$$\\frac{\\partial{^{4}} \\Delta^{{b_i}}}{{\\partial \\tau^{3}}{\\partial \\delta}} = b_{i} \\left(b_{i}^{3} \\frac{\\partial}{\\partial \\delta} \\Delta{\\left (\\tau,\\delta \\right )} \\left(\\frac{\\partial}{\\partial \\tau} \\Delta{\\left (\\tau,\\delta \\right )}\\right)^{3} + 3 b_{i}^{2} \\Delta{\\left (\\tau,\\delta \\right )} \\frac{\\partial}{\\partial \\delta} \\Delta{\\left (\\tau,\\delta \\right )} \\frac{\\partial}{\\partial \\tau} \\Delta{\\left (\\tau,\\delta \\right )} \\frac{\\partial^{2}}{\\partial \\tau^{2}} \\Delta{\\left (\\tau,\\delta \\right )} + 3 b_{i}^{2} \\Delta{\\left (\\tau,\\delta \\right )} \\left(\\frac{\\partial}{\\partial \\tau} \\Delta{\\left (\\tau,\\delta \\right )}\\right)^{2} \\frac{\\partial^{2}}{\\partial \\delta\\partial \\tau} \\Delta{\\left (\\tau,\\delta \\right )} - 6 b_{i}^{2} \\frac{\\partial}{\\partial \\delta} \\Delta{\\left (\\tau,\\delta \\right )} \\left(\\frac{\\partial}{\\partial \\tau} \\Delta{\\left (\\tau,\\delta \\right )}\\right)^{3} + b_{i} \\Delta^{2}{\\left (\\tau,\\delta \\right )} \\frac{\\partial}{\\partial \\delta} \\Delta{\\left (\\tau,\\delta \\right )} \\frac{\\partial^{3}}{\\partial \\tau^{3}} \\Delta{\\left (\\tau,\\delta \\right )} + 3 b_{i} \\Delta^{2}{\\left (\\tau,\\delta \\right )} \\frac{\\partial}{\\partial \\tau} \\Delta{\\left (\\tau,\\delta \\right )} \\frac{\\partial^{3}}{\\partial \\delta\\partial \\tau^{2}} \\Delta{\\left (\\tau,\\delta \\right )} + 3 b_{i} \\Delta^{2}{\\left (\\tau,\\delta \\right )} \\frac{\\partial^{2}}{\\partial \\delta\\partial \\tau} \\Delta{\\left (\\tau,\\delta \\right )} \\frac{\\partial^{2}}{\\partial \\tau^{2}} \\Delta{\\left (\\tau,\\delta \\right )} - 9 b_{i} \\Delta{\\left (\\tau,\\delta \\right )} \\frac{\\partial}{\\partial \\delta} \\Delta{\\left (\\tau,\\delta \\right )} \\frac{\\partial}{\\partial \\tau} \\Delta{\\left (\\tau,\\delta \\right )} \\frac{\\partial^{2}}{\\partial \\tau^{2}} \\Delta{\\left (\\tau,\\delta \\right )} - 9 b_{i} \\Delta{\\left (\\tau,\\delta \\right )} \\left(\\frac{\\partial}{\\partial \\tau} \\Delta{\\left (\\tau,\\delta \\right )}\\right)^{2} \\frac{\\partial^{2}}{\\partial \\delta\\partial \\tau} \\Delta{\\left (\\tau,\\delta \\right )} + 11 b_{i} \\frac{\\partial}{\\partial \\delta} \\Delta{\\left (\\tau,\\delta \\right )} \\left(\\frac{\\partial}{\\partial \\tau} \\Delta{\\left (\\tau,\\delta \\right )}\\right)^{3} + \\Delta^{3}{\\left (\\tau,\\delta \\right )} \\frac{\\partial^{4}}{\\partial \\delta\\partial \\tau^{3}} \\Delta{\\left (\\tau,\\delta \\right )} - \\Delta^{2}{\\left (\\tau,\\delta \\right )} \\frac{\\partial}{\\partial \\delta} \\Delta{\\left (\\tau,\\delta \\right )} \\frac{\\partial^{3}}{\\partial \\tau^{3}} \\Delta{\\left (\\tau,\\delta \\right )} - 3 \\Delta^{2}{\\left (\\tau,\\delta \\right )} \\frac{\\partial}{\\partial \\tau} \\Delta{\\left (\\tau,\\delta \\right )} \\frac{\\partial^{3}}{\\partial \\delta\\partial \\tau^{2}} \\Delta{\\left (\\tau,\\delta \\right )} - 3 \\Delta^{2}{\\left (\\tau,\\delta \\right )} \\frac{\\partial^{2}}{\\partial \\delta\\partial \\tau} \\Delta{\\left (\\tau,\\delta \\right )} \\frac{\\partial^{2}}{\\partial \\tau^{2}} \\Delta{\\left (\\tau,\\delta \\right )} + 6 \\Delta{\\left (\\tau,\\delta \\right )} \\frac{\\partial}{\\partial \\delta} \\Delta{\\left (\\tau,\\delta \\right )} \\frac{\\partial}{\\partial \\tau} \\Delta{\\left (\\tau,\\delta \\right )} \\frac{\\partial^{2}}{\\partial \\tau^{2}} \\Delta{\\left (\\tau,\\delta \\right )} + 6 \\Delta{\\left (\\tau,\\delta \\right )} \\left(\\frac{\\partial}{\\partial \\tau} \\Delta{\\left (\\tau,\\delta \\right )}\\right)^{2} \\frac{\\partial^{2}}{\\partial \\delta\\partial \\tau} \\Delta{\\left (\\tau,\\delta \\right )} - 6 \\frac{\\partial}{\\partial \\delta} \\Delta{\\left (\\tau,\\delta \\right )} \\left(\\frac{\\partial}{\\partial \\tau} \\Delta{\\left (\\tau,\\delta \\right )}\\right)^{3}\\right) \\Delta^{b_{i} - 4}{\\left (\\tau,\\delta \\right )}$$"
|
|
],
|
|
"metadata": {},
|
|
"output_type": "display_data",
|
|
"text": [
|
|
"<IPython.core.display.Math at 0xb9e4a20>"
|
|
]
|
|
},
|
|
{
|
|
"latex": [
|
|
"$$\\frac{\\partial{^{4}} \\Delta^{{b_i}}}{{\\partial \\tau^{2}}{\\partial \\delta^{2}}} = b_{i} \\left(b_{i}^{3} \\left(\\frac{\\partial}{\\partial \\delta} \\Delta{\\left (\\tau,\\delta \\right )}\\right)^{2} \\left(\\frac{\\partial}{\\partial \\tau} \\Delta{\\left (\\tau,\\delta \\right )}\\right)^{2} + b_{i}^{2} \\Delta{\\left (\\tau,\\delta \\right )} \\left(\\frac{\\partial}{\\partial \\delta} \\Delta{\\left (\\tau,\\delta \\right )}\\right)^{2} \\frac{\\partial^{2}}{\\partial \\tau^{2}} \\Delta{\\left (\\tau,\\delta \\right )} + 4 b_{i}^{2} \\Delta{\\left (\\tau,\\delta \\right )} \\frac{\\partial}{\\partial \\delta} \\Delta{\\left (\\tau,\\delta \\right )} \\frac{\\partial}{\\partial \\tau} \\Delta{\\left (\\tau,\\delta \\right )} \\frac{\\partial^{2}}{\\partial \\delta\\partial \\tau} \\Delta{\\left (\\tau,\\delta \\right )} + b_{i}^{2} \\Delta{\\left (\\tau,\\delta \\right )} \\left(\\frac{\\partial}{\\partial \\tau} \\Delta{\\left (\\tau,\\delta \\right )}\\right)^{2} \\frac{\\partial^{2}}{\\partial \\delta^{2}} \\Delta{\\left (\\tau,\\delta \\right )} - 6 b_{i}^{2} \\left(\\frac{\\partial}{\\partial \\delta} \\Delta{\\left (\\tau,\\delta \\right )}\\right)^{2} \\left(\\frac{\\partial}{\\partial \\tau} \\Delta{\\left (\\tau,\\delta \\right )}\\right)^{2} + 2 b_{i} \\Delta^{2}{\\left (\\tau,\\delta \\right )} \\frac{\\partial}{\\partial \\delta} \\Delta{\\left (\\tau,\\delta \\right )} \\frac{\\partial^{3}}{\\partial \\delta\\partial \\tau^{2}} \\Delta{\\left (\\tau,\\delta \\right )} + 2 b_{i} \\Delta^{2}{\\left (\\tau,\\delta \\right )} \\frac{\\partial}{\\partial \\tau} \\Delta{\\left (\\tau,\\delta \\right )} \\frac{\\partial^{3}}{\\partial \\delta^{2}\\partial \\tau} \\Delta{\\left (\\tau,\\delta \\right )} + b_{i} \\Delta^{2}{\\left (\\tau,\\delta \\right )} \\frac{\\partial^{2}}{\\partial \\delta^{2}} \\Delta{\\left (\\tau,\\delta \\right )} \\frac{\\partial^{2}}{\\partial \\tau^{2}} \\Delta{\\left (\\tau,\\delta \\right )} + 2 b_{i} \\Delta^{2}{\\left (\\tau,\\delta \\right )} \\left(\\frac{\\partial^{2}}{\\partial \\delta\\partial \\tau} \\Delta{\\left (\\tau,\\delta \\right )}\\right)^{2} - 3 b_{i} \\Delta{\\left (\\tau,\\delta \\right )} \\left(\\frac{\\partial}{\\partial \\delta} \\Delta{\\left (\\tau,\\delta \\right )}\\right)^{2} \\frac{\\partial^{2}}{\\partial \\tau^{2}} \\Delta{\\left (\\tau,\\delta \\right )} - 12 b_{i} \\Delta{\\left (\\tau,\\delta \\right )} \\frac{\\partial}{\\partial \\delta} \\Delta{\\left (\\tau,\\delta \\right )} \\frac{\\partial}{\\partial \\tau} \\Delta{\\left (\\tau,\\delta \\right )} \\frac{\\partial^{2}}{\\partial \\delta\\partial \\tau} \\Delta{\\left (\\tau,\\delta \\right )} - 3 b_{i} \\Delta{\\left (\\tau,\\delta \\right )} \\left(\\frac{\\partial}{\\partial \\tau} \\Delta{\\left (\\tau,\\delta \\right )}\\right)^{2} \\frac{\\partial^{2}}{\\partial \\delta^{2}} \\Delta{\\left (\\tau,\\delta \\right )} + 11 b_{i} \\left(\\frac{\\partial}{\\partial \\delta} \\Delta{\\left (\\tau,\\delta \\right )}\\right)^{2} \\left(\\frac{\\partial}{\\partial \\tau} \\Delta{\\left (\\tau,\\delta \\right )}\\right)^{2} + \\Delta^{3}{\\left (\\tau,\\delta \\right )} \\frac{\\partial^{4}}{\\partial \\delta^{2}\\partial \\tau^{2}} \\Delta{\\left (\\tau,\\delta \\right )} - 2 \\Delta^{2}{\\left (\\tau,\\delta \\right )} \\frac{\\partial}{\\partial \\delta} \\Delta{\\left (\\tau,\\delta \\right )} \\frac{\\partial^{3}}{\\partial \\delta\\partial \\tau^{2}} \\Delta{\\left (\\tau,\\delta \\right )} - 2 \\Delta^{2}{\\left (\\tau,\\delta \\right )} \\frac{\\partial}{\\partial \\tau} \\Delta{\\left (\\tau,\\delta \\right )} \\frac{\\partial^{3}}{\\partial \\delta^{2}\\partial \\tau} \\Delta{\\left (\\tau,\\delta \\right )} - \\Delta^{2}{\\left (\\tau,\\delta \\right )} \\frac{\\partial^{2}}{\\partial \\delta^{2}} \\Delta{\\left (\\tau,\\delta \\right )} \\frac{\\partial^{2}}{\\partial \\tau^{2}} \\Delta{\\left (\\tau,\\delta \\right )} - 2 \\Delta^{2}{\\left (\\tau,\\delta \\right )} \\left(\\frac{\\partial^{2}}{\\partial \\delta\\partial \\tau} \\Delta{\\left (\\tau,\\delta \\right )}\\right)^{2} + 2 \\Delta{\\left (\\tau,\\delta \\right )} \\left(\\frac{\\partial}{\\partial \\delta} \\Delta{\\left (\\tau,\\delta \\right )}\\right)^{2} \\frac{\\partial^{2}}{\\partial \\tau^{2}} \\Delta{\\left (\\tau,\\delta \\right )} + 8 \\Delta{\\left (\\tau,\\delta \\right )} \\frac{\\partial}{\\partial \\delta} \\Delta{\\left (\\tau,\\delta \\right )} \\frac{\\partial}{\\partial \\tau} \\Delta{\\left (\\tau,\\delta \\right )} \\frac{\\partial^{2}}{\\partial \\delta\\partial \\tau} \\Delta{\\left (\\tau,\\delta \\right )} + 2 \\Delta{\\left (\\tau,\\delta \\right )} \\left(\\frac{\\partial}{\\partial \\tau} \\Delta{\\left (\\tau,\\delta \\right )}\\right)^{2} \\frac{\\partial^{2}}{\\partial \\delta^{2}} \\Delta{\\left (\\tau,\\delta \\right )} - 6 \\left(\\frac{\\partial}{\\partial \\delta} \\Delta{\\left (\\tau,\\delta \\right )}\\right)^{2} \\left(\\frac{\\partial}{\\partial \\tau} \\Delta{\\left (\\tau,\\delta \\right )}\\right)^{2}\\right) \\Delta^{b_{i} - 4}{\\left (\\tau,\\delta \\right )}$$"
|
|
],
|
|
"metadata": {},
|
|
"output_type": "display_data",
|
|
"text": [
|
|
"<IPython.core.display.Math at 0xb8dbac8>"
|
|
]
|
|
},
|
|
{
|
|
"latex": [
|
|
"$$\\frac{\\partial{^{4}} \\Delta^{{b_i}}}{{\\partial \\tau}{\\partial \\delta^{3}}} = b_{i} \\left(\\left(b_{i}^{3} - 6 b_{i}^{2} + 11 b_{i} - 6\\right) \\left(\\frac{\\partial}{\\partial \\delta} \\Delta{\\left (\\tau,\\delta \\right )}\\right)^{3} \\frac{\\partial}{\\partial \\tau} \\Delta{\\left (\\tau,\\delta \\right )} + \\left(3 b_{i} \\frac{\\partial}{\\partial \\delta} \\Delta{\\left (\\tau,\\delta \\right )} \\frac{\\partial^{3}}{\\partial \\delta^{2}\\partial \\tau} \\Delta{\\left (\\tau,\\delta \\right )} + b_{i} \\frac{\\partial}{\\partial \\tau} \\Delta{\\left (\\tau,\\delta \\right )} \\frac{\\partial^{3}}{\\partial \\delta^{3}} \\Delta{\\left (\\tau,\\delta \\right )} + 3 b_{i} \\frac{\\partial^{2}}{\\partial \\delta^{2}} \\Delta{\\left (\\tau,\\delta \\right )} \\frac{\\partial^{2}}{\\partial \\delta\\partial \\tau} \\Delta{\\left (\\tau,\\delta \\right )} - 3 \\frac{\\partial}{\\partial \\delta} \\Delta{\\left (\\tau,\\delta \\right )} \\frac{\\partial^{3}}{\\partial \\delta^{2}\\partial \\tau} \\Delta{\\left (\\tau,\\delta \\right )} - \\frac{\\partial}{\\partial \\tau} \\Delta{\\left (\\tau,\\delta \\right )} \\frac{\\partial^{3}}{\\partial \\delta^{3}} \\Delta{\\left (\\tau,\\delta \\right )} - 3 \\frac{\\partial^{2}}{\\partial \\delta^{2}} \\Delta{\\left (\\tau,\\delta \\right )} \\frac{\\partial^{2}}{\\partial \\delta\\partial \\tau} \\Delta{\\left (\\tau,\\delta \\right )}\\right) \\Delta^{2}{\\left (\\tau,\\delta \\right )} + 3 \\left(b_{i}^{2} \\frac{\\partial}{\\partial \\delta} \\Delta{\\left (\\tau,\\delta \\right )} \\frac{\\partial^{2}}{\\partial \\delta\\partial \\tau} \\Delta{\\left (\\tau,\\delta \\right )} + b_{i}^{2} \\frac{\\partial}{\\partial \\tau} \\Delta{\\left (\\tau,\\delta \\right )} \\frac{\\partial^{2}}{\\partial \\delta^{2}} \\Delta{\\left (\\tau,\\delta \\right )} - 3 b_{i} \\frac{\\partial}{\\partial \\delta} \\Delta{\\left (\\tau,\\delta \\right )} \\frac{\\partial^{2}}{\\partial \\delta\\partial \\tau} \\Delta{\\left (\\tau,\\delta \\right )} - 3 b_{i} \\frac{\\partial}{\\partial \\tau} \\Delta{\\left (\\tau,\\delta \\right )} \\frac{\\partial^{2}}{\\partial \\delta^{2}} \\Delta{\\left (\\tau,\\delta \\right )} + 2 \\frac{\\partial}{\\partial \\delta} \\Delta{\\left (\\tau,\\delta \\right )} \\frac{\\partial^{2}}{\\partial \\delta\\partial \\tau} \\Delta{\\left (\\tau,\\delta \\right )} + 2 \\frac{\\partial}{\\partial \\tau} \\Delta{\\left (\\tau,\\delta \\right )} \\frac{\\partial^{2}}{\\partial \\delta^{2}} \\Delta{\\left (\\tau,\\delta \\right )}\\right) \\Delta{\\left (\\tau,\\delta \\right )} \\frac{\\partial}{\\partial \\delta} \\Delta{\\left (\\tau,\\delta \\right )} + \\Delta^{3}{\\left (\\tau,\\delta \\right )} \\frac{\\partial^{4}}{\\partial \\delta^{3}\\partial \\tau} \\Delta{\\left (\\tau,\\delta \\right )}\\right) \\Delta^{b_{i} - 4}{\\left (\\tau,\\delta \\right )}$$"
|
|
],
|
|
"metadata": {},
|
|
"output_type": "display_data",
|
|
"text": [
|
|
"<IPython.core.display.Math at 0xb8dbac8>"
|
|
]
|
|
},
|
|
{
|
|
"latex": [
|
|
"$$\\frac{\\partial{^{4}} \\Delta^{{b_i}}}{{}{\\partial \\delta^{4}}} = b_{i} \\left(6 \\left(b_{i}^{2} - 3 b_{i} + 2\\right) \\Delta{\\left (\\tau,\\delta \\right )} \\left(\\frac{\\partial}{\\partial \\delta} \\Delta{\\left (\\tau,\\delta \\right )}\\right)^{2} \\frac{\\partial^{2}}{\\partial \\delta^{2}} \\Delta{\\left (\\tau,\\delta \\right )} + \\left(b_{i}^{3} - 6 b_{i}^{2} + 11 b_{i} - 6\\right) \\left(\\frac{\\partial}{\\partial \\delta} \\Delta{\\left (\\tau,\\delta \\right )}\\right)^{4} + \\left(4 b_{i} \\frac{\\partial}{\\partial \\delta} \\Delta{\\left (\\tau,\\delta \\right )} \\frac{\\partial^{3}}{\\partial \\delta^{3}} \\Delta{\\left (\\tau,\\delta \\right )} + 3 b_{i} \\left(\\frac{\\partial^{2}}{\\partial \\delta^{2}} \\Delta{\\left (\\tau,\\delta \\right )}\\right)^{2} - 4 \\frac{\\partial}{\\partial \\delta} \\Delta{\\left (\\tau,\\delta \\right )} \\frac{\\partial^{3}}{\\partial \\delta^{3}} \\Delta{\\left (\\tau,\\delta \\right )} - 3 \\left(\\frac{\\partial^{2}}{\\partial \\delta^{2}} \\Delta{\\left (\\tau,\\delta \\right )}\\right)^{2}\\right) \\Delta^{2}{\\left (\\tau,\\delta \\right )} + \\Delta^{3}{\\left (\\tau,\\delta \\right )} \\frac{\\partial^{4}}{\\partial \\delta^{4}} \\Delta{\\left (\\tau,\\delta \\right )}\\right) \\Delta^{b_{i} - 4}{\\left (\\tau,\\delta \\right )}$$"
|
|
],
|
|
"metadata": {},
|
|
"output_type": "display_data",
|
|
"text": [
|
|
"<IPython.core.display.Math at 0xc7a26d8>"
|
|
]
|
|
}
|
|
],
|
|
"prompt_number": 10
|
|
},
|
|
{
|
|
"cell_type": "code",
|
|
"collapsed": false,
|
|
"input": [],
|
|
"language": "python",
|
|
"metadata": {},
|
|
"outputs": [],
|
|
"prompt_number": 9
|
|
}
|
|
],
|
|
"metadata": {}
|
|
}
|
|
]
|
|
} |