def idiff(eq, y, x, n=1):
    """Return ``dy/dx`` assuming that ``eq == 0``.

    Parameters
    ==========

    y : the dependent variable or a list of dependent variables (with y first)
    x : the variable that the derivative is being taken with respect to
    n : the order of the derivative (default is 1)

    Examples
    ========

    >>> from sympy.abc import x, y, a
    >>> from sympy.geometry.util import idiff

    >>> circ = x**2 + y**2 - 4
    >>> idiff(circ, y, x)
    -x/y
    >>> idiff(circ, y, x, 2).simplify()
    -(x**2 + y**2)/y**3

    Here, ``a`` is assumed to be independent of ``x``:

    >>> idiff(x + a + y, y, x)
    -1

    Now the x-dependence of ``a`` is made explicit by listing ``a`` after
    ``y`` in a list.

    >>> idiff(x + a + y, [y, a], x)
    -Derivative(a, x) - 1

    See Also
    ========

    sympy.core.function.Derivative: represents unevaluated derivatives
    sympy.core.function.diff: explicitly differentiates wrt symbols

    """
    if is_sequence(y):
        dep = set(y)
        y = y[0]
    elif isinstance(y, Symbol):
        dep = set([y])
    else:
        raise ValueError("expecting x-dependent symbol(s) but got: %s" % y)

    f = dict([(s, Function(
        s.name)(x)) for s in eq.atoms(Symbol) if s != x and s in dep])
    dydx = Function(y.name)(x).diff(x)
    eq = eq.subs(f)
    derivs = {}
    for i in range(n):
        yp = solve(eq.diff(x), dydx)[0].subs(derivs)
        if i == n - 1:
            return yp.subs([(v, k) for k, v in f.items()])
        derivs[dydx] = yp
        eq = dydx - yp
        dydx = dydx.diff(x)

 def _main(expr):
     _new = []
     for a in expr.args:
         is_V = False
         if isinstance(a, V):
             is_V = True
             a = a.expr
         if a.is_Derivative:
             variables = a.atoms()
             func = a.expr
             variables.add(func)
             name = a.expr.__class__.__name__
             if ',' in name:
                 a = Function('%s' % name +
                       ''.join(map(str, a.variables)))(*variables)
             else:
                 a = Function('%s' % name + ',' +
                       ''.join(map(str, a.variables)))(*variables)
         #TODO add more, maybe all that have args
         elif a.is_Add or a.is_Mul or a.is_Pow:
             a = _main(a)
         if is_V:
             a = V(a)
             a.function = func
         _new.append( a )
     return expr.func(*tuple(_new))

def test_latex_printer():
    r = Function('r')('t')
    assert VectorLatexPrinter().doprint(r ** 2) == "r^{2}"
    r2 = Function('r^2')('t')
    assert VectorLatexPrinter().doprint(r2.diff()) == r'\dot{r^{2}}'
    ra = Function('r__a')('t')
    assert VectorLatexPrinter().doprint(ra.diff().diff()) == r'\ddot{r^{a}}'

def test_noncommutative_issue_15131():
    x = Symbol('x', commutative=False)
    t = Symbol('t', commutative=False)
    fx = Function('Fx', commutative=False)(x)
    ft = Function('Ft', commutative=False)(t)
    A = Symbol('A', commutative=False)
    eq = fx * A * ft
    eqdt = eq.diff(t)
    assert eqdt.args[-1] == ft.diff(t)

def test_issue_7687():
    from sympy.core.function import Function
    from sympy.abc import x
    f = Function('f')(x)
    ff = Function('f')(x)
    match_with_cache = ff.matches(f)
    assert isinstance(f, type(ff))
    clear_cache()
    ff = Function('f')(x)
    assert isinstance(f, type(ff))
    assert match_with_cache == ff.matches(f)

def test_lambdify_Derivative_arg_issue_16468():
    f = Function('f')(x)
    fx = f.diff()
    assert lambdify((f, fx), f + fx)(10, 5) == 15
    assert eval(lambdastr((f, fx), f/fx))(10, 5) == 2
    raises(SyntaxError, lambda:
        eval(lambdastr((f, fx), f/fx, dummify=False)))
    assert eval(lambdastr((f, fx), f/fx, dummify=True))(10, 5) == 2
    assert eval(lambdastr((fx, f), f/fx, dummify=True))(10, 5) == S.Half
    assert lambdify(fx, 1 + fx)(41) == 42
    assert eval(lambdastr(fx, 1 + fx, dummify=True))(41) == 42

def test_find_simple_recurrence():
    a = Function('a')
    n = Symbol('n')
    assert find_simple_recurrence([fibonacci(k) for k in range(12)]) == (
        -a(n) - a(n + 1) + a(n + 2))

    f = Function('a')
    i = Symbol('n')
    a = [1, 1, 1]
    for k in range(15): a.append(5*a[-1]-3*a[-2]+8*a[-3])
    assert find_simple_recurrence(a, A=f, N=i) == (
        -8*f(i) + 3*f(i + 1) - 5*f(i + 2) + f(i + 3))
    assert find_simple_recurrence([0, 2, 15, 74, 12, 3, 0,
                                    1, 2, 85, 4, 5, 63]) == 0

def test_simple():
    sympy.var('x, y, r')
    u = Function('u')(x, y)
    w = Function('w')(x, y)
    f = Function('f')(x, y)
    e = (u.diff(x) + 1./2*w.diff(x,x)**2)*f.diff(x,y) \
            + w.diff(x,y)*f.diff(x,x)
    return Vexpr(e, u, w)

def test_solve_for_functions_derivatives():
    t = Symbol('t')
    x = Function('x')(t)
    y = Function('y')(t)
    a11,a12,a21,a22,b1,b2 = symbols('a11,a12,a21,a22,b1,b2')

    soln = solve([a11*x + a12*y - b1, a21*x + a22*y - b2], x, y)
    assert soln == {
        x : (a22*b1 - a12*b2)/(a11*a22 - a12*a21),
        y : (a11*b2 - a21*b1)/(a11*a22 - a12*a21),
        }

    assert solve(x - 1, x) == [1]
    assert solve(3*x - 2, x) == [Rational(2, 3)]

    soln = solve([a11*x.diff(t) + a12*y.diff(t) - b1, a21*x.diff(t) +
            a22*y.diff(t) - b2], x.diff(t), y.diff(t))
    assert soln == { y.diff(t) : (a11*b2 - a21*b1)/(a11*a22 - a12*a21),
            x.diff(t) : (a22*b1 - a12*b2)/(a11*a22 - a12*a21) }

    assert solve(x.diff(t)-1, x.diff(t)) == [1]
    assert solve(3*x.diff(t)-2, x.diff(t)) == [Rational(2,3)]

    eqns = set((3*x - 1, 2*y-4))
    assert solve(eqns, set((x,y))) == { x : Rational(1, 3), y: 2 }
    x = Symbol('x')
    f = Function('f')
    F = x**2 + f(x)**2 - 4*x - 1
    assert solve(F.diff(x), diff(f(x), x)) == [-((x - 2)/f(x))]

    # Mixed cased with a Symbol and a Function
    x = Symbol('x')
    y = Function('y')(t)

    soln = solve([a11*x + a12*y.diff(t) - b1, a21*x +
            a22*y.diff(t) - b2], x, y.diff(t))
    assert soln == { y.diff(t) : (a11*b2 - a21*b1)/(a11*a22 - a12*a21),
            x : (a22*b1 - a12*b2)/(a11*a22 - a12*a21) }

Lagrange formalism

@author: Topher
"""

from __future__ import division

import sympy as sp
from sympy import sin,cos,Function

t = sp.Symbol('t')
params = sp.symbols('M , G , J , J_ball , R')
M , G , J , J_ball , R = params

# ball position r
r_t = Function('r')(t)
d_r_t = r_t.diff(t)
dd_r_t = r_t.diff(t,2)
# beam angle theta
theta_t = Function('theta')(t)
d_theta_t = theta_t.diff(t)
dd_theta_t = theta_t.diff(t,2)
# torque of the beam
tau = Function('tau')

# kinetic energy
T = ((M + J_ball/R**2)*d_r_t**2 + (J + M*r_t**2 + J_ball)*d_theta_t**2)/2

# potential energy
V = M*G*r_t*sin(theta_t)

def _test_f():
    # FIXME: we get infinite recursion here:
    f = Function("f")
    assert residue(f(x)/x**5, x, 0) == f.diff(x, 4)/24

import sys
import numpy as np
import sympy as sm
from sympy.solvers import solve
from sympy import Symbol, Function
import GUI as gui
import math
GCodeFile = open('C:\\3D Printer Calculus Project\\docs\\Sample.txt', 'w+')
x = Symbol('x')
f = Function('f')(x)
f = 0.5*x
xVals = []
yVals = []
zVals = []
SideArray = []
TotalVolume = 0
AxisChoice = gui.AxisRevScreen()
if AxisChoice == "X-axis":
	Bounds = gui.BoundsScreen()
	FirstBound = int(Bounds[0])
	FinalBound = int(Bounds[1])
	xInitialVal = FirstBound
	yInitialVal = f.subs(x, FirstBound)
	zInitialVal = 0
	xVals.append(xInitialVal)
	yVals.append(yInitialVal)
	zVals.append(zInitialVal)
	FinalBound1 = int(FinalBound * 10)
	FirstBound1 = int(FirstBound * 10)
	for t in range (FirstBound1, FinalBound1):
	### This for loop generates the x coordinates ###

from numpy import array, arange
from sympy import symbols, Function, S, solve, simplify, \
        collect, Matrix, lambdify

from pydy import ReferenceFrame, cross, dot, dt, express, expression2vector, \
    coeff

m, g, r1, r2, t = symbols("m, g r1 r2 t")
au1, au2, au3 = symbols("au1 au2 au3")
cf1, cf2, cf3 = symbols("cf1 cf2 cf3")
I, J = symbols("I J")
u3p, u4p, u5p = symbols("u3p u4p u5p")

q1 = Function("q1")(t)
q2 = Function("q2")(t)
q3 = Function("q3")(t)
q4 = Function("q4")(t)
q5 = Function("q5")(t)

def eval(a):
    subs_dict = {
        u3.diff(t): u3p,
        u4.diff(t): u4p,
        u5.diff(t): u5p,
        r1:1,
        r2:0,
        m:1,
        g:1,
        I:1,
        J:1,
        }

print "r =", g2

#Propagation constants along the axes are related
g3 = alpha1**2 + beta1**2 + gamma1**2

# Simplifying
g3=g3.subs(sin(phi)**2, 1-cos(phi)**2).expand().simplify()

print r'%\alpha^{2} + \beta^{2} + \gamma^{2} = ', latex(g3)

x_hat = Matrix([
    rho * sin(theta) * cos(phi),
    rho * sin(theta) * sin(phi),
    rho * cos(theta)])

psi = Function("psi")

psi =exp(I*omega*t) * exp(-I*g2)
print "\Psi =", latex(psi)

#Derivatives of the wave function of the coordinates
dpsidx=psi.diff(x)
dpsidx=dpsidx.subs(psi,'Psi').expand().simplify()
print "d \Psi / dx =", latex(dpsidx)

dpsidy=psi.diff(y)
dpsidy=dpsidy.subs(psi,'Psi').expand().simplify()
print "d \Psi / dy =", latex(dpsidy)

dpsidz=psi.diff(z)
dpsidz=dpsidz.subs(psi,'Psi').expand().simplify()

def test_cylinder_clpt():
    '''Test case where the functional corresponds to the internal energy of
    a cylinder using the Classical Laminated Plate Theory (CLPT)

    '''
    from sympy import Matrix

    sympy.var('x, y, r')
    sympy.var('B11, B12, B16, B21, B22, B26, B61, B62, B66')
    sympy.var('D11, D12, D16, D21, D22, D26, D61, D62, D66')

    # displacement field
    u = Function('u')(x, y)
    v = Function('v')(x, y)
    w = Function('w')(x, y)
    # stress function
    f = Function('f')(x, y)
    # laminate constitute matrices B represents B*, see Jones (1999)
    B = Matrix([[B11, B12, B16],
                [B21, B22, B26],
                [B61, B62, B66]])
    # D represents D*, see Jones (1999)
    D = Matrix([[D11, D12, D16],
                [D12, D22, D26],
                [D16, D26, D66]])
    # strain-diplacement equations
    e = Matrix([[u.diff(x) + 1./2*w.diff(x)**2],
                [v.diff(y) + 1./r*w + 1./2*w.diff(y)**2],
                [u.diff(y) + v.diff(x) + w.diff(x)*w.diff(y)]])
    k = Matrix([[  -w.diff(x, x)],
                [  -w.diff(y, y)],
                [-2*w.diff(x, y)]])
    # representing the internal forces using the stress function
    N = Matrix([[  f.diff(y, y)],
                [  f.diff(x, x)],
                [ -f.diff(x, y)]])
    functional = N.T*V(e) - N.T*B*V(k) + k.T*D.T*V(k)
    return Vexpr(functional, u, v, w)

        ICD11 ICD22 ICD33 ICD13 IEF11 IEF22 IEF33 IEF13 IF22 g')
# Declare coordinates and their time derivatives
q, qd = N.declare_coords('q', 11)
# Declare speeds and their time derivatives
u, ud = N.declare_speeds('u', 6)
# Unpack the lists
rr, rrt, rf, rft, lr, ls, lf, l1, l2, l3, l4, mcd, mef, IC22, ICD11, ICD22,\
        ICD33, ICD13, IEF11, IEF22, IEF33, IEF13, IF22, g = params
q1, q2, q3, q4, q5, q6, q7, q8, q9, q10, q11 = q
q1d, q2d, q3d, q4d, q5d, q6d, q7d, q8d, q9d, q10d, q11d = qd
u1, u2, u3, u4, u5, u6 = u
u1d, u2d, u3d, u4d, u5d, u6d = ud

tan_lean = {sin(q2)/cos(q2): tan(q2)}
# Create variables for to act as place holders in the vector from FO to FN
g31 = Function('g31')(t)
g33 = Function('g33')(t)
g31_s, g33_s = symbols('g31 g33')
g31d_s, g33d_s = symbols('g31d g33d')

"""
# Some simplifying symbols / trig expressions
s1, s2, s3, s4, s5, s6, c1, c2, c3, c4, c5, c6, t2 = symbols('s1 \
        s2 s3 s4 s5 s6 c1 c2 c3 c4 c5 c6 t2')

symbol_subs_dict = {sin(q1): s1,
                    cos(q1): c1,
                    tan(q2): t2,
                    cos(q2): c2,
                    sin(q2): s2,
                    cos(q3): c3,

var("x y")
d_u = Symbol("D_u")
d_v = Symbol("D_v")
sigma = Symbol("SIGMA")
lam = Symbol("LAMBDA")
kappa = Symbol("KAPPA")
k = Symbol("K")

def eq1(u, v, f):
    return -d_u*d_u*(u.diff(x, 2) + u.diff(y, 2)) - f(u) + sigma*v

def eq2(u, v):
    return -d_v*d_v*(v.diff(x, 2) + v.diff(y, 2)) - u + v

u = Function("u")
v = Function("v")
print "Reaction-Diffusion Equations:"
def f(u):
    return lam*u-u**3-kappa
pprint(eq1(u(x, y), v(x, y), f))
pprint(eq2(u(x, y), v(x, y)))

print "\nSolution:"
u_hat = 1 - (exp(k*x)+exp(-k*x)) / (exp(k)+exp(-k))
pprint(Eq(Function("u_hat")(x), u_hat))

#test the Boundary Conditions:
assert u_hat.subs(x, -1) == 0
assert u_hat.subs(x, 1) == 0

from sympy import Function, Symbol, symbols, Derivative, preview, simplify, collect, Wild
from sympy import diff, ln, sin, pprint, sqrt, latex, Integral
import sympy as sym

x, y, z, t = symbols('x y z t')
f = Function('f')
F = Function('F')(x, y, z)
g = Function('g', real=True)(ln(F))

result = diff(sin(x), x)

print(result)
print(f(x * x).diff(x))
print(g.diff(x))

aaa = g.diff(x, 2)
# aaa._args[1] * aaa._args[0]
# init_printing()

pprint(Integral(sqrt(1 / x), x), use_unicode=True)
print(latex(Integral(sqrt(1 / x), x)))

print("\n\n")
pprint(g.diff((x, 1), (y, 0)), use_unicode=True)
# pprint(g.diff((x, 2),(y, 2)), use_unicode=True)

pprint(sym.diff(sym.tan(x), x))
pprint(sym.diff(g, x))

print("xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx")
from sympy import Derivative as D, collect, Function