r"""
Calculus Tests and Examples.

Compute the Christoffel symbol.

::

    sage: var('r t theta phi')
    (r, t, theta, phi)
    sage: m = matrix(SR, [[(1-1/r),0,0,0],[0,-(1-1/r)^(-1),0,0],[0,0,-r^2,0],[0,0,0,-r^2*(sin(theta))^2]])
    sage: print m
    [         -1/r + 1                 0                 0                 0]
    [                0       1/(1/r - 1)                 0                 0]
    [                0                 0              -r^2                 0]
    [                0                 0                 0 -r^2*sin(theta)^2]

::

    sage: def christoffel(i,j,k,vars,g):
    ...   s = 0
    ...   ginv = g^(-1)
    ...   for l in range(g.nrows()):
    ...      s = s + (1/2)*ginv[k,l]*(g[j,l].diff(vars[i])+g[i,l].diff(vars[j])-g[i,j].diff(vars[l]))
    ...   return s

::

    sage: christoffel(3,3,2, [t,r,theta,phi], m)
    -sin(theta)*cos(theta)
    sage: X = christoffel(1,1,1,[t,r,theta,phi],m)
    sage: X
     1/2/((1/r - 1)*r^2)
    sage: X.rational_simplify()
     -1/2/(r^2 - r)

Some basic things::

    sage: f(x,y) = x^3 + sinh(1/y)                   
    sage: f
    (x, y) |--> x^3 + sinh(1/y)
    sage: f^3
    (x, y) |--> (x^3 + sinh(1/y))^3
    sage: (f^3).expand()
    (x, y) |--> x^9 + 3*x^6*sinh(1/y) + 3*x^3*sinh(1/y)^2 + sinh(1/y)^3

A polynomial over a symbolic base ring::

    sage: R = SR[x]
    sage: f = R([1/sqrt(2), 1/(4*sqrt(2))])
    sage: f
    1/8*sqrt(2)*x + 1/2*sqrt(2)    
    sage: -f
    -1/8*sqrt(2)*x - 1/2*sqrt(2)    
    sage: (-f).degree()
    1

A big product.  Notice that simplifying simplifies the product further::

    sage: A = exp(I*pi/5)       
    sage: b = A*A*A*A*A*A*A*A*A*A
    sage: b
    1

We check a statement made at the beginning of Friedlander and
Joshi's book on Distributions::

    sage: f(x) = sin(x^2)
    sage: g(x) = cos(x) + x^3
    sage: u = f(x+t) + g(x-t)
    sage: u
    -(t - x)^3 + sin((t + x)^2) + cos(-t + x)
    sage: u.diff(t,2) - u.diff(x,2)
    0

Restoring variables after they have been turned into functions::

    sage: x = function('x')
    sage: type(x)
    <class 'sage.symbolic.function_factory.NewSymbolicFunction'>
    sage: x(2/3)
    x(2/3)
    sage: restore('x')
    sage: sin(x).variables()
    (x,)

MATHEMATICA: Some examples of integration and differentiation taken
from some Mathematica docs::

    sage: var('x n a')
    (x, n, a)
    sage: diff(x^n, x)        # the output looks funny, but is correct
    n*x^(n - 1)
    sage: diff(x^2 * log(x+a), x)
    2*x*log(a + x) + x^2/(a + x)
    sage: derivative(arctan(x), x)
    1/(x^2 + 1)
    sage: derivative(x^n, x, 3)
    (n - 2)*(n - 1)*n*x^(n - 3)
    sage: derivative( function('f')(x), x)
    D[0](f)(x)    
    sage: diff( 2*x*f(x^2), x)
    4*x^2*D[0](f)(x^2) + 2*f(x^2)
    sage: integrate( 1/(x^4 - a^4), x)
    1/4*log(-a + x)/a^3 - 1/4*log(a + x)/a^3 - 1/2*arctan(x/a)/a^3
    sage: expand(integrate(log(1-x^2), x))
    x*log(-x^2 + 1) - 2*x - log(x - 1) + log(x + 1)
    sage: integrate(log(1-x^2)/x, x)
    1/2*log(-x^2 + 1)*log(x^2) + 1/2*polylog(2, -x^2 + 1)
    sage: integrate(exp(1-x^2),x)
    1/2*sqrt(pi)*e*erf(x)
    sage: integrate(sin(x^2),x)
    1/8*((I - 1)*sqrt(2)*erf((1/2*I - 1/2)*sqrt(2)*x) + (I + 1)*sqrt(2)*erf((1/2*I + 1/2)*sqrt(2)*x))*sqrt(pi)    

    sage: integrate((1-x^2)^n,x)
    integrate((-x^2 + 1)^n, x)
    sage: integrate(x^x,x)
    integrate(x^x, x)
    sage: integrate(1/(x^3+1),x)
    1/3*sqrt(3)*arctan(1/3*(2*x - 1)*sqrt(3)) + 1/3*log(x + 1) - 1/6*log(x^2 - x + 1)
    sage: integrate(1/(x^3+1), x, 0, 1)
    1/9*pi*sqrt(3) + 1/3*log(2)

::

    sage: forget()
    sage: c = var('c')
    sage: assume(c > 0)
    sage: integrate(exp(-c*x^2), x, -oo, oo)
    sqrt(pi)/sqrt(c)
    sage: forget()

The following are a bunch of examples of integrals that Mathematica
can do, but Sage currently can't do::

    sage: integrate(sqrt(x + sqrt(x)), x)    # todo -- Mathematica can do this
    integrate(sqrt(x + sqrt(x)), x)
    sage: integrate(log(x)*exp(-x^2), x)        # todo -- Mathematica can do this
    integrate(e^(-x^2)*log(x), x)

Todo - Mathematica can do this and gets `\pi^2/15`.

::

    sage: integrate(log(1+sqrt(1+4*x)/2)/x, x, 0, 1)  # not tested
    Traceback (most recent call last):
    ...
    ValueError: Integral is divergent.

::

    sage: integrate(ceil(x^2 + floor(x)), x, 0, 5)    # todo: Mathematica can do this
    integrate(ceil(x^2) + floor(x), x, 0, 5)

MAPLE: The basic differentiation and integration examples in the
Maple documentation::

    sage: diff(sin(x), x)
    cos(x)
    sage: diff(sin(x), y)
    0
    sage: diff(sin(x), x, 3)
    -cos(x)
    sage: diff(x*sin(cos(x)), x)
    -x*sin(x)*cos(cos(x)) + sin(cos(x))
    sage: diff(tan(x), x)
    tan(x)^2 + 1
    sage: f = function('f'); f
    f
    sage: diff(f(x), x)
    D[0](f)(x)    
    sage: diff(f(x,y), x, y)
    D[0, 1](f)(x, y)    
    sage: diff(f(x,y), x, y) - diff(f(x,y), y, x)
    0
    sage: g = function('g')
    sage: var('x y z')
    (x, y, z)
    sage: diff(g(x,y,z), x,z,z)
    D[0, 2, 2](g)(x, y, z)    
    sage: integrate(sin(x), x)
    -cos(x)
    sage: integrate(sin(x), x, 0, pi)
    2

::

    sage: var('a b')
    (a, b)
    sage: integrate(sin(x), x, a, b)
    cos(a) - cos(b)

::

    sage: integrate( x/(x^3-1), x)
    1/3*sqrt(3)*arctan(1/3*(2*x + 1)*sqrt(3)) + 1/3*log(x - 1) - 1/6*log(x^2 + x + 1)
    sage: integrate(exp(-x^2), x)
    1/2*sqrt(pi)*erf(x)    
    sage: integrate(exp(-x^2)*log(x), x)       # todo: maple can compute this exactly.
    integrate(e^(-x^2)*log(x), x)
    sage: f = exp(-x^2)*log(x)
    sage: f.nintegral(x, 0, 999)
    (-0.87005772672831..., 7.5584...e-10, 567, 0)
    sage: integral(1/sqrt(2*t^4 - 3*t^2 - 2), t, 2, 3)     # todo: maple can do this
    integrate(1/sqrt(2*t^4 - 3*t^2 - 2), t, 2, 3)
    sage: integral(integral(x*y^2, x, 0, y), y, -2, 2)
    32/5

We verify several standard differentiation rules::

    sage: function('f, g')
    (f, g)
    sage: diff(f(t)*g(t),t)
    f(t)*D[0](g)(t) + g(t)*D[0](f)(t)
    sage: diff(f(t)/g(t), t)
    -f(t)*D[0](g)(t)/g(t)^2 + D[0](f)(t)/g(t)
    sage: diff(f(t) + g(t), t)
    D[0](f)(t) + D[0](g)(t)    
    sage: diff(c*f(t), t)
    c*D[0](f)(t)    
"""