Calculus Tests and Examples - Symbolic Calculus

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# Calculus Tests and Examples

Compute the Christoffel symbol.

Sage

    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: 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]

Python

    >>> from sage.all import *
    >>> var('r t theta phi')
    (r, t, theta, phi)
    >>> m = matrix(SR, [[(Integer(1)-Integer(1)/r),Integer(0),Integer(0),Integer(0)],[Integer(0),-(Integer(1)-Integer(1)/r)**(-Integer(1)),Integer(0),Integer(0)],[Integer(0),Integer(0),-r**Integer(2),Integer(0)],[Integer(0),Integer(0),Integer(0),-r**Integer(2)*(sin(theta))**Integer(2)]])
    >>> 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

    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

[Python]

    >>> from sage.all import *
    >>> def christoffel(i, j, k, vars, g):
    ...     s = Integer(0)
    ...     ginv = g**(-Integer(1))
    ...     for l in range(g.nrows()):
    ...         s = s + (Integer(1)/Integer(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

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

Python

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

Some basic things:

Sage

    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

[Python]

    >>> from sage.all import *
    >>> __tmp__=var("x,y"); f = symbolic_expression(x**Integer(3) + sinh(Integer(1)/y)).function(x,y)
    >>> f
    (x, y) |--> x^3 + sinh(1/y)
    >>> f**Integer(3)
    (x, y) |--> (x^3 + sinh(1/y))^3
    >>> (f**Integer(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

    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

Python

    >>> from sage.all import *
    >>> R = SR['x']
    >>> f = R([Integer(1)/sqrt(Integer(2)), Integer(1)/(Integer(4)*sqrt(Integer(2)))])
    >>> f
    1/8*sqrt(2)*x + 1/2*sqrt(2)
    >>> -f
    -1/8*sqrt(2)*x - 1/2*sqrt(2)
    >>> (-f).degree()
    1

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

Sage

    sage: A = exp(I*pi/7)
    sage: b = A^14
    sage: b
    1

[Python]

    >>> from sage.all import *
    >>> A = exp(I*pi/Integer(7))
    >>> b = A**Integer(14)
    >>> b
    1

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

Sage

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

Python

    >>> from sage.all import *
    >>> __tmp__=var("x"); f = symbolic_expression(sin(x**Integer(2))).function(x)
    >>> __tmp__=var("x"); g = symbolic_expression(cos(x) + x**Integer(3)).function(x)
    >>> u = f(x+t) + g(x-t)
    >>> u
    -(t - x)^3 + cos(-t + x) + sin((t + x)^2)
    >>> u.diff(t,Integer(2)) - u.diff(x,Integer(2))
    0

Restoring variables after they have been turned into functions:

Sage

    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,)

[Python]

    >>> from sage.all import *
    >>> x = function('x')
    >>> type(x)
    <class 'sage.symbolic.function_factory...NewSymbolicFunction'>
    >>> x(Integer(2)/Integer(3))
    x(2/3)
    >>> restore('x')
    >>> sin(x).variables()
    (x,)

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

Sage

    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 - 1)*(n - 2)*n*x^(n - 3)
    sage: derivative( function('f')(x), x)
    diff(f(x), 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)
    -1/2*arctan(x/a)/a^3 - 1/4*log(a + x)/a^3 + 1/4*log(-a + x)/a^3
    sage: expand(integrate(log(1-x^2), x))
    x*log(-x^2 + 1) - 2*x + log(x + 1) - log(x - 1)

Python

    >>> from sage.all import *
    >>> var('x n a')
    (x, n, a)
    >>> diff(x**n, x)        # the output looks funny, but is correct
    n*x^(n - 1)
    >>> diff(x**Integer(2) * log(x+a), x)
    2*x*log(a + x) + x^2/(a + x)
    >>> derivative(arctan(x), x)
    1/(x^2 + 1)
    >>> derivative(x**n, x, Integer(3))
    (n - 1)*(n - 2)*n*x^(n - 3)
    >>> derivative( function('f')(x), x)
    diff(f(x), x)
    >>> diff( Integer(2)*x*f(x**Integer(2)), x)
    4*x^2*D[0](f)(x^2) + 2*f(x^2)
    >>> integrate( Integer(1)/(x**Integer(4) - a**Integer(4)), x)
    -1/2*arctan(x/a)/a^3 - 1/4*log(a + x)/a^3 + 1/4*log(-a + x)/a^3
    >>> expand(integrate(log(Integer(1)-x**Integer(2)), x))
    x*log(-x^2 + 1) - 2*x + log(x + 1) - log(x - 1)

This is an apparent regression in Maxima 5.39.0, although the antiderivative is correct, assuming we work with (poly)logs of complex argument. More convenient form is 1/2*log(x^2)*log(-x^2 + 1) + 1/2*dilog(-x^2 + 1). See also https://sourceforge.net/p/maxima/bugs/3275/:

Sage

    sage: integrate(log(1-x^2)/x, x)
    log(-x)*log(x + 1) + log(x)*log(-x + 1) + dilog(x + 1) + dilog(-x + 1)

[Python]

    >>> from sage.all import *
    >>> integrate(log(Integer(1)-x**Integer(2))/x, x)
    log(-x)*log(x + 1) + log(x)*log(-x + 1) + dilog(x + 1) + dilog(-x + 1)

No problems here:

Sage

    sage: integrate(exp(1-x^2),x)
    1/2*sqrt(pi)*erf(x)*e
    sage: integrate(sin(x^2),x)
    1/16*sqrt(pi)*((I + 1)*sqrt(2)*erf((1/2*I + 1/2)*sqrt(2)*x) - (I - 1)*sqrt(2)*erf(sqrt(-I)*x) - 2*sqrt(2)*imag_part(erf((-1)^(1/4)*x)) + 2*sqrt(2)*real_part(erf((-1)^(1/4)*x)))

    sage: integrate((1-x^2)^n,x)  # long time
    x*hypergeometric((1/2, -n), (3/2,), x^2*exp_polar(2*I*pi))
    sage: integrate(x^x,x)
    integrate(x^x, x)
    sage: integrate(1/(x^3+1),x)
    1/3*sqrt(3)*arctan(1/3*sqrt(3)*(2*x - 1)) - 1/6*log(x^2 - x + 1) + 1/3*log(x + 1)
    sage: integrate(1/(x^3+1), x, 0, 1)
    1/9*sqrt(3)*pi + 1/3*log(2)

Python

    >>> from sage.all import *
    >>> integrate(exp(Integer(1)-x**Integer(2)),x)
    1/2*sqrt(pi)*erf(x)*e
    >>> integrate(sin(x**Integer(2)),x)
    1/16*sqrt(pi)*((I + 1)*sqrt(2)*erf((1/2*I + 1/2)*sqrt(2)*x) - (I - 1)*sqrt(2)*erf(sqrt(-I)*x) - 2*sqrt(2)*imag_part(erf((-1)^(1/4)*x)) + 2*sqrt(2)*real_part(erf((-1)^(1/4)*x)))

    >>> integrate((Integer(1)-x**Integer(2))**n,x)  # long time
    x*hypergeometric((1/2, -n), (3/2,), x^2*exp_polar(2*I*pi))
    >>> integrate(x**x,x)
    integrate(x^x, x)
    >>> integrate(Integer(1)/(x**Integer(3)+Integer(1)),x)
    1/3*sqrt(3)*arctan(1/3*sqrt(3)*(2*x - 1)) - 1/6*log(x^2 - x + 1) + 1/3*log(x + 1)
    >>> integrate(Integer(1)/(x**Integer(3)+Integer(1)), x, Integer(0), Integer(1))
    1/9*sqrt(3)*pi + 1/3*log(2)

Sage

    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()

[Python]

    >>> from sage.all import *
    >>> forget()
    >>> c = var('c')
    >>> assume(c > Integer(0))
    >>> integrate(exp(-c*x**Integer(2)), x, -oo, oo)
    sqrt(pi)/sqrt(c)
    >>> forget()

Other examples that now (Issue #27958) work:

Sage

    sage: integrate(log(x)*exp(-x^2), x)  # long time
    1/2*sqrt(pi)*erf(x)*log(x) - x*hypergeometric((1/2, 1/2), (3/2, 3/2), -x^2)

    sage: integrate(log(1+sqrt(1+4*x)/2)/x, x, 0, 1)
    Traceback (most recent call last):
    ...
    ValueError: Integral is divergent.

Python

    >>> from sage.all import *
    >>> integrate(log(x)*exp(-x**Integer(2)), x)  # long time
    1/2*sqrt(pi)*erf(x)*log(x) - x*hypergeometric((1/2, 1/2), (3/2, 3/2), -x^2)

    >>> integrate(log(Integer(1)+sqrt(Integer(1)+Integer(4)*x)/Integer(2))/x, x, Integer(0), Integer(1))
    Traceback (most recent call last):
    ...
    ValueError: Integral is divergent.

The following is an example of integral that Mathematica can do, but Sage currently cannot do:

Sage

    sage: integrate(ceil(x^2 + floor(x)), x, 0, 5, algorithm='maxima')
    integrate(ceil(x^2) + floor(x), x, 0, 5)

[Python]

    >>> from sage.all import *
    >>> integrate(ceil(x**Integer(2) + floor(x)), x, Integer(0), Integer(5), algorithm='maxima')
    integrate(ceil(x^2) + floor(x), x, 0, 5)

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

Sage

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

Python

    >>> from sage.all import *
    >>> diff(sin(x), x)
    cos(x)
    >>> diff(sin(x), y)
    0
    >>> diff(sin(x), x, Integer(3))
    -cos(x)
    >>> diff(x*sin(cos(x)), x)
    -x*cos(cos(x))*sin(x) + sin(cos(x))
    >>> diff(tan(x), x)
    tan(x)^2 + 1
    >>> f = function('f'); f
    f
    >>> diff(f(x), x)
    diff(f(x), x)
    >>> diff(f(x,y), x, y)
    diff(f(x, y), x, y)
    >>> diff(f(x,y), x, y) - diff(f(x,y), y, x)
    0
    >>> g = function('g')
    >>> var('x y z')
    (x, y, z)
    >>> diff(g(x,y,z), x,z,z)
    diff(g(x, y, z), x, z, z)
    >>> integrate(sin(x), x)
    -cos(x)
    >>> integrate(sin(x), x, Integer(0), pi)
    2

Sage

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

[Python]

    >>> from sage.all import *
    >>> var('a b')
    (a, b)
    >>> integrate(sin(x), x, a, b)
    cos(a) - cos(b)

Sage

    sage: integrate( x/(x^3-1), x)
    1/3*sqrt(3)*arctan(1/3*sqrt(3)*(2*x + 1)) - 1/6*log(x^2 + x + 1) + 1/3*log(x - 1)
    sage: integrate(exp(-x^2), x)
    1/2*sqrt(pi)*erf(x)
    sage: integrate(exp(-x^2)*log(x), x)  # long time
    1/2*sqrt(pi)*erf(x)*log(x) - x*hypergeometric((1/2, 1/2), (3/2, 3/2), -x^2)
    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)     # long time  # todo: maple can do this
    integrate(1/(sqrt(2*t^2 + 1)*sqrt(t^2 - 2)), t, 2, 3)
    sage: integral(integral(x*y^2, x, 0, y), y, -2, 2)
    32/5

Python

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

We verify several standard differentiation rules:

Sage

    sage: function('f, g')
    (f, g)
    sage: diff(f(t)*g(t),t)
    g(t)*diff(f(t), t) + f(t)*diff(g(t), t)
    sage: diff(f(t)/g(t), t)
    diff(f(t), t)/g(t) - f(t)*diff(g(t), t)/g(t)^2
    sage: diff(f(t) + g(t), t)
    diff(f(t), t) + diff(g(t), t)
    sage: diff(c*f(t), t)
    c*diff(f(t), t)

[Python]

    >>> from sage.all import *
    >>> function('f, g')
    (f, g)
    >>> diff(f(t)*g(t),t)
    g(t)*diff(f(t), t) + f(t)*diff(g(t), t)
    >>> diff(f(t)/g(t), t)
    diff(f(t), t)/g(t) - f(t)*diff(g(t), t)/g(t)^2
    >>> diff(f(t) + g(t), t)
    diff(f(t), t) + diff(g(t), t)
    >>> diff(c*f(t), t)
    c*diff(f(t), t)