Taylor Expansion with 2 variables

[hamorabi]

I'm sharing a little function I wrote that returns the Taylor expansion of functions of two variables. The Taylor function included in the prime only works for single variable functions.

Function definition:

PHP Code:

taylor2d(ff,vars,exppt,ordr) 


For example, to return the second order polynomial that approximates the function sin(2x)+cos(y) at the point x=0, y=0:

PHP Code:

taylor2d(sin(2x)+cos(y),[x,y],[0,0],2) 


You can also use this function to find the equation of the tangential plane to a surface. Since the tangential plane is a first order approximation, just set the parameter ordr to 1:

PHP Code:

taylor2d(ln(2*x+y),[x,y],[-1,3],1) 


will give the answer:

PHP Code:

2x+y-1 


Equate the result to 0 and you have the equation of the tangential plane.

PHP Code:

#cas
def taylor2d(ff,vars,exppt,ordr):
  """
  taylor2d(sin(2x)+cos(y),[x,y],[0,0],2)
  taylor2d(x*e^y+1,[x,y],[1,0],2)
  """
  if (type(vars) != DOM_LIST) and (type(exppt) != DOM_LIST):
    return taylor(ff,equal(vars,exppt),ordr,polynom)
  elif (len(vars) == 2 and len(exppt) == 2):
    pp = seq(equal(vars[ii],exppt[ii]),ii,0,len(vars)-1)
    ii, nn, bb, dd = 0, 0, 0, 0
    while (ii<=ordr):
      bb = binomial(ii,nn)/(ii!)*((vars[0]-exppt[0])**nn)*((vars[1]-exppt[1])**(ii-nn))
      dd += subst(diff(diff(ff,vars[0],nn),vars[1],ii-nn),pp)*bb
      if (nn==0):
        ii += 1
        nn = ii + 1
      nn -= 1
    return dd
  else:
    return "Error!"
#end