I am developing an improved API for differentiation in sympy which is a
class for scalar differential operators (a linear combination of sympy
expressions and partial derivatives). The code (test_dop.py attached)
produces the following output -
[image: test_dop-1.jpg]
To run this code also requires the gprinter.py and dop.py modules (also
attached). Note that the scalar differential operator API could be
extended to vector operators or differential forms. It only depends on how
you define the coefficients of the partial derivative operators. You could
also define partial derivative operators that operate on the left argument
instead of on the right argument of *.
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#gprinter.py
import sys,shutil,os
from sympy import init_printing, latex
try:
from IPython.display import display, Latex, Math, display_latex
except ImportError:
pass
try:
from sympy.interactive import printing
except ImportError:
pass
from sympy import *
def isinteractive(): #Is ipython running
'''
We will assume that if ipython is running then jupyter notebook is
running.
'''
try:
__IPYTHON__
return True
except NameError:
return False
class gprinter:
'''
The gprinter class implements the functions needed to print sympy
objects in a function, gprint, that mimics the syntax
of the python 3 print function. Additionally, it also impliments latex
printing from Jupyter Notebook/Lab and from python scripts. For latex
printing from linux and windows the following programs are required:
Jupyter Notebook/Lab is sufficient for printing latex in both linux and
windows since in both cases it now includes Mathjax.
To print latex from a python script in linux a tex distribution is
required. It is suggested that the texlive-full distribution which
includes pdfcrop be installed with the command:
sudo apt-get install texlive-full
To print latex from a python script in windows 10 the full texlive
installation is also suggested. Use the following link to download
the windows texlive installation program:
http://mirror.ctan.org/systems/texlive/tlnet/install-tl-windows.exe
In order to use the pdfcrop program in texlive a perl distribution must
be installed. Here is a suggested installation link:
http://strawberryperl.com/
'''
line_sep = \
"""
************************************************************************
"""
latex_preamble = \
"""
\\pagestyle{empty}
\\usepackage[latin1]{inputenc}
\\usepackage{amsmath}
\\usepackage{amsfonts}
\\usepackage{amssymb}
\\usepackage{amsbsy}
\\usepackage{tensor}
\\usepackage{listings}
\\usepackage{color}
\\usepackage{xcolor}
\\usepackage{bm}
\\usepackage{breqn}
\\definecolor{gray}{rgb}{0.95,0.95,0.95}
\\setlength{\\parindent}{0pt}
\\DeclareMathOperator{\\Tr}{Tr}
\\DeclareMathOperator{\\Adj}{Adj}
\\newcommand{\\bfrac}[2]{\\displaystyle\\frac{#1}{#2}}
\\newcommand{\\bc}{\\begin{center}}
\\newcommand{\\ec}{\\end{center}}
\\newcommand{\\lp}{\\left (}
\\newcommand{\\rp}{\\right )}
\\newcommand{\\paren}[1]{\\lp {#1} \\rp}
\\newcommand{\\half}{\\frac{1}{2}}
\\newcommand{\\llt}{\\left <}
\\newcommand{\\rgt}{\\right >}
\\newcommand{\\abs}[1]{\\left |{#1}\\right | }
\\newcommand{\\pdiff}[2]{\\bfrac{\\partial {#1}}{\\partial {#2}}}
\\newcommand{\\lbrc}{\\left \\{}
\\newcommand{\\rbrc}{\\right \\}}
\\newcommand{\\W}{\\wedge}
\\newcommand{\\prm}[1]{{#1}'}
\\newcommand{\\ddt}[1]{\\bfrac{d{#1}}{dt}}
\\newcommand{\\R}{\\dagger}
\\newcommand{\\deriv}[3]{\\bfrac{d^{#3}#1}{d{#2}^{#3}}}
\\newcommand{\\grade}[2]{\\left < {#1} \\right >_{#2}}
\\newcommand{\\f}[2]{{#1}\\lp{#2}\\rp}
\\newcommand{\\eval}[2]{\\left . {#1} \\right |_{#2}}
\\newcommand{\\Nabla}{\\boldsymbol{\\nabla}}
\\newcommand{\\eb}{\\boldsymbol{e}}
\\newcommand{\\bs}[1]{\\boldsymbol{#1}}
\\newcommand{\\grad}{\\bs{\\nabla}}
\\usepackage{float}
\\floatstyle{plain}
\\newfloat{Code}{H}{myc}
\\lstloadlanguages{Python}
"""
ip_cmds = \
[r'$$\DeclareMathOperator{\Tr}{Tr}$$',\
r'$$\DeclareMathOperator{\Adj}{Adj}$$',\
r'$$\DeclareMathOperator{\sinc}{sinc}$$',\
r'$$\newcommand{\bfrac}[2]{\displaystyle\frac{#1}{#2}}$$',\
r'$$\newcommand{\lp}{\left (}$$',\
r'$$\newcommand{\rp}{\right )}$$',\
r'$$\newcommand{\paren}[1]{\lp {#1} \rp}$$',\
r'$$\newcommand{\half}{\frac{1}{2}}$$',\
r'$$\newcommand{\llt}{\left <}$$',\
r'$$\newcommand{\rgt}{\right >}$$',\
r'$$\newcommand{\abs}[1]{\left |{#1}\right | }$$',\
r'$$\newcommand{\pdiff}[2]{\bfrac{\partial {#1}}{\partial {#2}}}$$',\
r'$$\newcommand{\npdiff}[3]{\bfrac{\partial^{#3} {#1}}{\partial {#2}^{#3}}}$$',\
r'$$\newcommand{\lbrc}{\left \{}$$',\
r'$$\newcommand{\rbrc}{\right \}}$$',\
r'$$\newcommand{\W}{\wedge}$$',\
r'$$\newcommand{\prm}[1]{{#1}}$$',\
r'$$\newcommand{\ddt}[1]{\bfrac{d{#1}}{dt}}$$',\
r'$$\newcommand{\R}{\dagger}$$',\
r'$$\newcommand{\deriv}[3]{\bfrac{d^{#3}#1}{d{#2}^{#3}}}$$',\
r'$$\newcommand{\grade}[2]{\left < {#1} \right >_{#2}}$$',\
r'$$\newcommand{\f}[2]{{#1}\lp {#2} \rp}$$',\
r'$$\newcommand{\eval}[2]{\left . {#1} \right |_{#2}}$$',\
r'$$\newcommand{\bs}[1]{\boldsymbol{#1}}$$',\
r'$$\newcommand{\grad}{\bs{\nabla}}$$']
#***********************************************************************
SYS_CMD = {'linux2': {'rm': 'rm', 'evince': 'evince', 'null': ' > /dev/null', '&': '&'},
'linux': {'rm': 'rm', 'evince': 'evince', 'null': ' > /dev/null', '&': '&'},
'win32': {'rm': 'del', 'evince': 'start', 'null': ' > NUL', '&': ''},
'darwin': {'rm': 'rm', 'evince': 'open', 'null': ' > /dev/null', '&': '&'}}
latex_flg = False
latex_str = ''
Format_cnt = 0
pdiff_format = True
@classmethod
def format(cls,pdiff_format=True):
cls.pdiff_format = pdiff_format
if cls.Format_cnt == 0:
cls.Format_cnt += 1
cls.latex_flg = True # Latex printing for scripts or Notebooks
if isinteractive(): # Latex printing for Notebooks
init_printing(use_latex='mathjax')
from IPython.display import Math, display
cmds = '\n'.join(cls.ip_cmds)
display(Math(cmds))
return
@classmethod
def gprint(cls,*xargs):
x = []
fstr = ''
new_eq_flg = False
i = 0
for xi in xargs:
if isinstance(xi,str):
if r'\\' in xi and i > 0:
if isinteractive(): # Required for Jupyter Notebook/Lab
xi_rep = xi.replace(r'\\',r'\end{equation*}@\begin{equation*} ')
else: # Required for latex output from python scripts
xi_rep = xi.replace(r'\\',r'\end{equation*}'+'\n'+r'\begin{equation*} ')
new_eq_flg = True
fstr += xi_rep
else: # Pure text printing
fstr += xi
elif isinstance(xi,type): # Special case for printing python type of an object
if cls.latex_flg:
fstr += r' \text{'+str(xi)+'} '
else:
fstr += str(xi)
else:
if cls.latex_flg: # Convert object to latex string and append to printing string
x.append(latex(xi))
if new_eq_flg:
new_eq_flg = False
fstr += r' %s '
else: # Append text object to printing string
x.append(str(xi))
fstr += r' %s '
i += 1
if cls.latex_flg:
if isinteractive(): # Print all xargs to Jupyter Notebook/Lab
lstr = fstr % tuple(x)
if '@' in lstr:
lines = lstr.split('@')
lines[0] = r'\begin{equation*} '+lines[0]
lines[-1] += r'\end{equation*}'
for line in lines:
display(Math(line))
else:
display(Math(lstr))
else: # Add latex representatin of all xargs to total latex string
latex_str = (fstr % tuple(x))
if latex_str[0] == '#':
cls.latex_str += latex_str[1:]
else:
cls.latex_str += r'\begin{equation*} ' + latex_str + r'\end{equation*} '+'\n'
else: # Print all xargs in text mode
print(fstr % tuple(x))
return
@classmethod
def pdf(cls,filename=None, paper=(14, 11), crop=None, prog=False, debug=False, pt='10pt', pdfprog='pdflatex'):
'''
Post processes LaTeX output (see comments below), adds preamble and
postamble, generates tex file, inputs file to latex, displays resulting
pdf file.
Arg Value Result
pdfprog 'pdflatex' Use pdfprog to generate pdf output, only generates tex file if pdfprog is None
crop margin in bp Use "pdfcrop" to crop output file (pdfcrop must be installed, linux only)
bp is the TeX big point approximately 1/72 inch.
png True Use "convert" to produce png output (imagemagick must be installed, linux only)
We assume that if gprinter.pdf() is called then gprinter.Format() has been called at the beginning of the program.
'''
if not cls.latex_flg:
print('gprinter.Format() has not been called. No LaTeX string to process')
return
latex_str = cls.paper_format(paper,pt)+cls.latex_preamble+r'\begin{document}'+'\n'+cls.latex_str+r'\end{document}'
pyfilepath = sys.path[0]
if filename is None:
pyfilename = sys.argv[0]
rootfilename = pyfilename.replace('.py', '')
tex_filename = rootfilename + '.tex'
pdf_filename = rootfilename + '.pdf'
else:
tex_filename = filename
pdf_filename = tex_filname.replace('.tex','.pdf')
if debug:
print('latex file =', filename)
latex_file = open(tex_filename, 'w')
latex_file.write(latex_str)
latex_file.close()
sys_cmd = cls.SYS_CMD[sys.platform]
if pdfprog is not None:
pdflatex = shutil.which(pdfprog)
if debug: # Display latex excution output for debugging purposes
print('pdflatex path =', pdflatex)
else: # Works for Linux don't know about Windows
os.system(pdfprog+' '+tex_filename+sys_cmd['null'])
if debug:
pass
os.remove(os.path.join(pyfilepath,rootfilename+'.aux'))
os.remove(os.path.join(pyfilepath,rootfilename+'.log'))
else:
os.remove(os.path.join(pyfilepath,rootfilename+'.aux'))
os.remove(os.path.join(pyfilepath,rootfilename+'.log'))
os.remove(os.path.join(pyfilepath,rootfilename+'.tex'))
if crop is not None:
crop_cmd = 'pdfcrop --margins "'+str(crop)+'" '+pdf_filename
os.system(crop_cmd)
os.remove(os.path.join(pyfilepath,pdf_filename))
os.rename(os.path.join(rootfilename+'-crop.pdf'),os.path.join(pdf_filename))
os.system(sys_cmd['evince']+' '+pdf_filename)
return
@classmethod
def paper_format(cls,paper,pt): #Set size of paper and font size
if paper == 'letter':
paper_size = \
"""
\\documentclass[@10pt@,fleqn]{book}
"""
else:
paper_size = \
"""
\\documentclass[@10pt@,fleqn]{book}
\\usepackage[vcentering]{geometry}
"""
if paper == 'landscape':
paper = [11, 8.5]
paper_size += '\\geometry{papersize={' + str(paper[0]) + \
'in,' + str(paper[1]) + 'in},total={' + str(paper[0] - 1) + \
'in,' + str(paper[1] - 1) + 'in}}\n'
paper_size = paper_size.replace('@10pt@', pt)
return(paper_size)
def gprint(*args): #Shortcut so you don't have to code gprinter.gprint
gprinter.gprint(*args)
return
"""
Differential operators, for all sympy expressions
"""
import copy
import numbers
import warnings
from typing import List, Tuple, Any, Iterable
from sympy import Symbol, S, Add, simplify, diff, Expr, Dummy, expand,\
latex, Basic
from gprinter import gprinter
def apply_function_list(f, x):
if isinstance(f, (tuple, list)):
fx = x
for fi in f:
fx = fi(fx)
return fx
else:
return f(x)
class _BaseDop(Basic):
""" Base class for differential operators - used to avoid accidental promotion """
pass
#################### Partial Derivative Operator Class #################
def _basic_diff(f, x, n=1):
""" Simple wrapper for `diff` that works for our types too """
if isinstance(f, (Expr, Symbol, numbers.Number)): # f is sympy expression
return diff(f, x, n)
elif hasattr(f, '_eval_derivative_n_times'):
# one of our types
return f._eval_derivative_n_times(x, n)
else:
raise ValueError('In_basic_diff type(arg) = ' + str(type(f)) + ' not allowed.')
class Pdop(_BaseDop):
r"""
Partial derivative operatorp.
The partial derivatives are of the form
.. math::
\partial_{i_{1}...i_{n}} =
\frac{\partial^{i_{1}+...+i_{n}}}{\partial{x_{1}^{i_{1}}}...\partial{x_{n}^{i_{n}}}}.
If :math:`i_{j} = 0` then the partial derivative does not contain the
:math:`x^{i_{j}}` coordinate.
Attributes
----------
pdiffs : dict
A dictionary where coordinates are keys and key value are the number of
times one differentiates with respect to the key.
order : int
Total number of differentiations.
When this is zero (i.e. when :attr:`pdiffs` is ``{}``) then this object
is the identity operator, and returns its operand unchanged.
"""
pdiff_format = gprinter.pdiff_format
x = Symbol('x',real=True)
def sort_key(self, order=None):
return (
# lower order derivatives first
self.order,
# sorted by symbol after that, after expansion
sorted([
x.sort_key(order)
for x, k in self.pdiffs.items()
for i in range(k)
])
)
def __eq__(self, A):
if isinstance(A, Pdop) and self.pdiffs == A.pdiffs:
return True
else:
if len(self.pdiffs) == 0 and A == S.One:
return True
return False
def __init__(self, __arg):
"""
The partial differential operator is a partial derivative with
respect to a set of real symbols (variables).
"""
if __arg is None:
warnings.warn(
"`Pdop(None)` is deprecated, use `Pdop({})` instead",
DeprecationWarning, stacklevel=2)
__arg = {}
if isinstance(__arg, dict): # Pdop defined by dictionary
if len(__arg) == 0:
self.pdiffs = {Pdop.x:0}
else:
self.pdiffs = __arg
elif isinstance(__arg, Symbol): # First order derivative with respect to symbol
self.pdiffs = {__arg: 1}
else:
raise TypeError('A dictionary or symbol is required, got {!r}'.format(__arg))
self.order = sum(self.pdiffs.values())
def _eval_derivative_n_times(self, x, n) -> 'Pdop': # pdiff(self)
# d is partial derivative
pdiffs = copy.copy(self.pdiffs)
if x in pdiffs:
pdiffs[x] += n
else:
pdiffs[x] = n
return Pdop(pdiffs)
def __call__(self, arg):
"""
Calculate nth order partial derivative (order defined by
self) of expression
"""
for x, n in self.pdiffs.items():
arg = _basic_diff(arg, x, n)
return arg
def __pow__(self,other):
if isinstance(other,int):
p = self
for i in range(1,other):
p = p*self
return p
raise TypeError('For power of Pdop an integer is required, got {!r}'.format(other))
def __mul__(self, other): # functional product of self and arg (self*arg)
return self(other)
def __rmul__(self, other): # functional product of arg and self (arg*self)
assert not isinstance(other, Pdop)
return Sdop([(other, self)])
def __add__(self, other):
if isinstance(other,Pdop):
pd1 = Sdop([1],[self])
pd2 = Sdop([1],[other])
return pd1+pd2
elif isinstance(other,Sdop):
pd1 = Sdop([1],[self])
return pd1+other
else:
return Sdop([other],[Pdop({})])+Sdop([1],[self])
def __radd__(self, other):
return self+other
def _sympystr(self,printer):
if self.order == 0:
return 'D{}'
s = ''
for x in self.pdiffs:
n = self.pdiffs[x]
s += 'D'
if n > 1:
s += '^' + printer._print(n)
s += '{' + printer._print(x) + '}'
return s
def _latex(self,printer):
if self.order == 0:
return ''
if gprinter.pdiff_format:
s = r'\frac{\partial'
if self.order > 1:
s += '^{' + printer._print(self.order) + '}'
s += '}{'
keys = list(self.pdiffs.keys())
keys.sort(key=lambda x: x.sort_key())
for key in keys:
i = self.pdiffs[key]
s += r'\partial ' + printer._print(key)
if i > 1:
s += '^{' + printer._print(i) + '}'
s += '}'
else:
s = ''
keys = list(self.pdiffs.keys())
keys.sort(key=lambda x: x.sort_key())
for key in keys:
i = self.pdiffs[key]
s+= r'\partial'
if ( i > 1 ):
s += r'^{' + printer._print(i) + r'}'
s += r'_{' + printer._print(key) + '}'
return s
########################################################################
def _merge_terms(terms1, terms2):
""" Concatenate and consolidate two sets of already-consolidated terms """
pdiffs1 = [pdiff for _, pdiff in terms1]
pdiffs2 = [pdiff for _, pdiff in terms2]
pdiffs = pdiffs1 + [x for x in pdiffs2 if x not in pdiffs1]
coefs = len(pdiffs) * [S.Zero]
for coef, pdiff in terms1:
index = pdiffs.index(pdiff)
coefs[index] += coef
for coef, pdiff in terms2:
index = pdiffs.index(pdiff)
coefs[index] += coef
# remove zeros
return [(coef, pdiff) for coef, pdiff in zip(coefs, pdiffs) if coef != S.Zero]
def _eval_derivative_n_times_terms(terms, x, n):
for i in range(n):
new_terms = []
for k, term in enumerate(terms):
dc = _basic_diff(term[0], x)
pd = _basic_diff(term[1], x)
# print 'D0, term, dc, pd =', D0, term, dc, pd
if dc != 0:
new_terms.append((dc, term[1]))
if pd != 0:
new_terms.append((term[0], pd))
terms = new_terms
return _consolidate_terms(terms)
def _consolidate_terms(terms):
"""
Remove zero coefs and consolidate coefs with repeated pdiffs.
"""
new_coefs = []
new_pdiffs = []
for coef, pd in terms:
if coef != S.Zero:
if pd in new_pdiffs:
index = new_pdiffs.index(pd)
new_coefs[index] += coef
else:
new_coefs.append(coef)
new_pdiffs.append(pd)
return tuple(zip(new_coefs, new_pdiffs))
################ Scalar Partial Differential Operator Class ############
class Sdop(_BaseDop):
"""
Scalar differential operator is of the form (Einstein summation)
.. math:: D = c_{i}*D_{i}
where the :math:`c_{i}`'s are scalar coefficient (they could be functions)
and the :math:`D_{i}`'s are partial differential operators (:class:`Pdop`).
Attributes
----------
terms : tuple of tuple
the structure :math:`((c_{1},D_{1}),(c_{2},D_{2}), ...)`
"""
def TSimplify(self):
return Sdop([
(simplify(coef), pdiff) for coef, pdiff in self.terms
])
@staticmethod
def consolidate_coefs(sdop):
"""
Remove zero coefs and consolidate coefs with repeated pdiffs.
"""
if isinstance(sdop, Sdop):
return Sdop(_consolidate_terms(sdop.terms))
else:
return _consolidate_terms(sdop)
def simplify(self, modes=simplify):
return Sdop([
(metric.apply_function_list(modes, coef), pdiff)
for coef, pdiff in self.terms
])
def _with_sorted_terms(self):
new_terms = sorted(self.terms, key=lambda term: Pdop.sort_key(term[1]))
return Sdop(new_terms)
def _sympystr(self, print_obj):
if len(self.terms) == 0:
return ZERO_STR
self = self._with_sorted_terms()
s = ''
for coef, pdop in self.terms:
coef_str = print_obj._print(coef)
pd_str = print_obj._print(pdop)
if coef == S.One:
s += pd_str
elif coef == S.NegativeOne:
s += '-' + pd_str
else:
if isinstance(coef, Add):
s += '(' + coef_str + ')*' + pd_str
else:
s += coef_str + '*' + pd_str
s += ' + '
s = s.replace('+ -', '- ')
s = s[:-3]
return s
def _latex(self, printer):
if len(self.terms) == 0:
return ZERO_STR
self = self._with_sorted_terms()
s = ''
for coef, pdop in self.terms:
coef_str = printer._print(coef)
pd_str = printer._print(pdop)
if coef == S.One:
if pd_str == '':
s += '1'
else:
s += pd_str
elif coef == S.NegativeOne:
if pd_str == '':
s += '-1'
else:
s += '-' + pd_str
else:
if isinstance(coef, Add):
s += r'\left ( ' + coef_str + r'\right ) ' + pd_str
else:
s += coef_str + ' ' + pd_str
s += ' + '
s = s.replace('+ -', '- ')
return s[:-3]
def __init_from_symbol(self, symbol: Symbol) -> None:
self.terms = ((S.One, Pdop(symbol)),)
def __init_from_coef_and_pdiffs(self, coefs: List[Any], pdiffs: List['Pdop']) -> None:
if not isinstance(coefs, list) or not isinstance(pdiffs, list):
raise TypeError("coefs and pdiffs must be lists")
if len(coefs) != len(pdiffs):
raise ValueError('In Sdop.__init__ coefficent list and Pdop list must be same length.')
self.terms = tuple(zip(coefs, pdiffs))
def __init_from_terms(self, terms: Iterable[Tuple[Any, 'Pdop']]) -> None:
self.terms = tuple(terms)
def __init__(self, *args):
if len(args) == 1:
if isinstance(args[0], Symbol):
self.__init_from_symbol(*args)
elif isinstance(args[0], (list, tuple)):
self.__init_from_terms(*args)
else:
raise TypeError(
"A symbol or sequence is required (got type {})"
.format(type(args[0]).__name__))
elif len(args) == 2:
self.__init_from_coef_and_pdiffs(*args)
else:
raise TypeError(
"Sdop() takes from 1 to 2 positional arguments but {} were "
"given".format(len(args)))
def __call__(self, arg):
# Ensure that we return the right type even when there are no terms - we
# do this by adding `0 * d(arg)/d(nonexistant)`, which must be zero, but
# will be a zero of the right type.
dummy_var = Dummy('nonexistant')
terms = self.terms or ((S.Zero, Pdop(dummy_var)),)
return sum([coef * pdiff(arg) for coef, pdiff in terms])
def __neg__(self):
return Sdop([(-coef, pdiff) for coef, pdiff in self.terms])
@staticmethod
def Add(sdop1, sdop2):
if isinstance(sdop1, Sdop) and isinstance(sdop2, Sdop):
return Sdop(_merge_terms(sdop1.terms, sdop2.terms))
else:
# convert values to multiplicative operators
if not isinstance(sdop2, _BaseDop):
sdop2 = Sdop([(sdop2, Pdop({}))])
elif not isinstance(sdop1, _BaseDop):
sdop1 = Sdop([(sdop1, Pdop({}))])
else:
return NotImplemented
return Sdop.Add(sdop1, sdop2)
def __eq__(self, other):
if isinstance(other, Sdop):
diff = self - other
return len(diff.terms) == 0
else:
return NotImplemented
def __add__(self, other):
if isinstance(other, Pdop):
other = Sdop([1],[other])
elif isinstance(other, Expr):
other = Sdop([other],[Pdop({})])
else:
pass
return Sdop.Add(self, other)
def __pow__(self, other):
if isinstance(other,int):
p = self
for i in range(1,other):
p = p*self
return p
raise TypeError('For power of Sdop an integer is required, got {!r}'.format(other))
def __radd__(self, sdop):
if isinstance(sdop,Pdop):
sdop = Sdop([1],[sdop])
return Sdop.Add(sdop, self)
def __sub__(self, sdop):
return Sdop.Add(self, -sdop)
def __rsub__(self, sdop):
return Sdop.Add(-self, sdop)
def __mul__(self, sdopr):
# alias for applying the operator
return self.__call__(sdopr)
def __rmul__(self, sdop):
return Sdop([(sdop * coef, pdiff) for coef, pdiff in self.terms])
def _eval_derivative_n_times(self, x, n):
return Sdop(_eval_derivative_n_times_terms(self.terms, x, n))
from sympy import symbols
from gprinter import *
from dop import *
gprinter.format()
(x,y,z) = symbols('x y z',real=True)
r2 = x**2+y**2+z**2
Dx = Pdop(x)
Dy = Pdop(y)
Dz = Pdop(z)
nabla2 = Dx**2+Dy**2+Dz**2
gprint('r^2 = ',r2)
gprint(r'\nabla^2 = ',nabla2)
gprint(r'\nabla^2 r^2 = ',nabla2*r2)
gprint(r'r^2 \nabla^2 = ',r2*nabla2)
gprint(r'r^2 \nabla^2 r^2 = ',r2*nabla2*r2)
gprint(r'r^2+ \nabla^2 = ',r2+nabla2)
rdotgrad = x*Dx+y*Dy+z*Dz
gprint(r'\bs{r}\cdot\bs{\nabla} = ',rdotgrad)
gprint(r'\bs{r}\cdot\bs{\nabla} r^2= ',rdotgrad*r2)
gprinter.pdf(crop=5)
