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: 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]
>>> 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]
var('r t theta phi')
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]])
m
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
>>> 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
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
>>> 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
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)
-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)
>>> 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)
christoffel(3,3,2, [t,r,theta,phi], m) X = christoffel(1,1,1,[t,r,theta,phi],m) X X.rational_simplify()
>>> 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)
christoffel(3,3,2, [t,r,theta,phi], m) X = christoffel(1,1,1,[t,r,theta,phi],m) X X.rational_simplify()
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
>>> 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
f(x,y) = x^3 + sinh(1/y) f f^3 (f^3).expand()
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
>>> 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
R = SR['x'] f = R([1/sqrt(2), 1/(4*sqrt(2))]) f -f (-f).degree()
A big product. Notice that simplifying simplifies the product further:
sage: A = exp(I*pi/7)
sage: b = A^14
sage: b
1
>>> from sage.all import *
>>> A = exp(I*pi/Integer(7))
>>> b = A**Integer(14)
>>> b
1
A = exp(I*pi/7) b = A^14 b
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 + cos(-t + x) + sin((t + x)^2)
sage: u.diff(t,2) - u.diff(x,2)
0
>>> 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
f(x) = sin(x^2) g(x) = cos(x) + x^3 u = f(x+t) + g(x-t) u u.diff(t,2) - u.diff(x,2)
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,)
>>> 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,)
x = function('x')
type(x)
x(2/3)
restore('x')
sin(x).variables()
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 - 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)
>>> 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)
var('x n a')
diff(x^n, x) # the output looks funny, but is correct
diff(x^2 * log(x+a), x)
derivative(arctan(x), x)
derivative(x^n, x, 3)
derivative( function('f')(x), x)
diff( 2*x*f(x^2), x)
integrate( 1/(x^4 - a^4), x)
expand(integrate(log(1-x^2), x))
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: integrate(log(1-x^2)/x, x)
log(-x)*log(x + 1) + log(x)*log(-x + 1) + dilog(x + 1) + dilog(-x + 1)
>>> 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)
integrate(log(1-x^2)/x, x)
No problems here:
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((1/2*I - 1/2)*sqrt(2)*x) - (I - 1)*sqrt(2)*erf(sqrt(-I)*x) + (I + 1)*sqrt(2)*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)
>>> 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((1/2*I - 1/2)*sqrt(2)*x) - (I - 1)*sqrt(2)*erf(sqrt(-I)*x) + (I + 1)*sqrt(2)*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)
integrate(exp(1-x^2),x) integrate(sin(x^2),x) integrate((1-x^2)^n,x) # long time integrate(x^x,x) integrate(1/(x^3+1),x) integrate(1/(x^3+1), x, 0, 1)
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()
>>> 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()
forget()
c = var('c')
assume(c > 0)
integrate(exp(-c*x^2), x, -oo, oo)
forget()
>>> 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()
forget()
c = var('c')
assume(c > 0)
integrate(exp(-c*x^2), x, -oo, oo)
forget()
Other examples that now (Issue #27958) work:
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.
>>> 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.
integrate(log(x)*exp(-x^2), x) # long time integrate(log(1+sqrt(1+4*x)/2)/x, x, 0, 1)
The following is an example of integral that Mathematica can do, but Sage currently cannot do:
sage: integrate(ceil(x^2 + floor(x)), x, 0, 5, algorithm='maxima')
integrate(ceil(x^2) + floor(x), x, 0, 5)
>>> 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)
integrate(ceil(x^2 + floor(x)), x, 0, 5, algorithm='maxima')
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*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
>>> 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
diff(sin(x), x)
diff(sin(x), y)
diff(sin(x), x, 3)
diff(x*sin(cos(x)), x)
diff(tan(x), x)
f = function('f'); f
diff(f(x), x)
diff(f(x,y), x, y)
diff(f(x,y), x, y) - diff(f(x,y), y, x)
g = function('g')
var('x y z')
diff(g(x,y,z), x,z,z)
integrate(sin(x), x)
integrate(sin(x), x, 0, pi)
sage: var('a b')
(a, b)
sage: integrate(sin(x), x, a, b)
cos(a) - cos(b)
>>> from sage.all import *
>>> var('a b')
(a, b)
>>> integrate(sin(x), x, a, b)
cos(a) - cos(b)
var('a b')
integrate(sin(x), x, a, b)
>>> from sage.all import *
>>> var('a b')
(a, b)
>>> integrate(sin(x), x, a, b)
cos(a) - cos(b)
var('a b')
integrate(sin(x), x, a, b)
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
>>> 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
integrate( x/(x^3-1), x) integrate(exp(-x^2), x) integrate(exp(-x^2)*log(x), x) # long time f = exp(-x^2)*log(x) f.nintegral(x, 0, 999) integral(1/sqrt(2*t^4 - 3*t^2 - 2), t, 2, 3) # long time # todo: maple can do this integral(integral(x*y^2, x, 0, y), y, -2, 2)
>>> 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
integrate( x/(x^3-1), x) integrate(exp(-x^2), x) integrate(exp(-x^2)*log(x), x) # long time f = exp(-x^2)*log(x) f.nintegral(x, 0, 999) integral(1/sqrt(2*t^4 - 3*t^2 - 2), t, 2, 3) # long time # todo: maple can do this integral(integral(x*y^2, x, 0, y), y, -2, 2)
We verify several standard differentiation rules:
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)
>>> 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)
function('f, g')
diff(f(t)*g(t),t)
diff(f(t)/g(t), t)
diff(f(t) + g(t), t)
diff(c*f(t), t)