Coverage for local/lib/python2.7/site-packages/sage/calculus/tests.py : 100%

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: 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) 

-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) 

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/7) 

sage: b = A^14 

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 + cos(-t + x) + sin((t + x)^2) 

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 - 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) 

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) 

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) 

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*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) 

:: 

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(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) 

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*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 

:: 

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*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) # 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) 

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) 

"""