Sympy (Symbolic Python)

The equation $$\int_a^b f(x)\,dx = \lim_{n\rightarrow\infty} \sum_{i=1}^n f(x_i^*)\Delta x$$ defines the definite integral over the interval $[a,b]$.

In [16]:

from sympy import init_session
init_session()

IPython console for SymPy 0.7.5 (Python 2.7.6-64-bit) (ground types: python)

These commands were executed:
>>> from __future__ import division
>>> from sympy import *
>>> x, y, z, t = symbols('x y z t')
>>> k, m, n = symbols('k m n', integer=True)
>>> f, g, h = symbols('f g h', cls=Function)

Documentation can be found at http://www.sympy.org

WARNING: Hook shutdown_hook is deprecated. Use the atexit module instead.

In [4]:

Integral(x**2/(x+1),(x,0,5))

Out[4]:

$$\int_{0}^{5} \frac{x^{2}}{x + 1}\, dx$$

In [5]:

simplify(sin(x)**2 + cos(x)**2)

Out[5]:

$$1$$

In [7]:

simplify((x**3 + x**2 - x - 1)/(x**2 + 2*x + 1))

Out[7]:

$$x - 1$$

In [8]:

simplify(gamma(x)/gamma(x - 2))

Out[8]:

$$\left(x - 2\right) \left(x - 1\right)$$

In [9]:

simplify(x**2 + 2*x + 1)

Out[9]:

$$x^{2} + 2 x + 1$$

In [10]:

factor(x**3 - 3*x**2 + 2*x - 5)

Out[10]:

$$x^{3} - 3 x^{2} + 2 x - 5$$

In [6]:

expr = (x - 2)*(2*x + 3)*(3*x - 4)*(7*x + 9)

In [7]:

expand(expr)

Out[7]:

$$42 x^{4} - 23 x^{3} - 197 x^{2} + 42 x + 216$$

In [8]:

factor(expand(expr))

Out[8]:

$$\left(x - 2\right) \left(2 x + 3\right) \left(3 x - 4\right) \left(7 x + 9\right)$$

In [14]:

expand((x + 1)*(x - 2) - (x - 1)*x)

Out[14]:

$$-2$$

In [9]:

factor(x**3 - x**2 + x - 1)

Out[9]:

$$\left(x - 1\right) \left(x^{2} + 1\right)$$

In [16]:

factor(x**2*z + 4*x*y*z + 4*y**2*z)

Out[16]:

$$z \left(x + 2 y\right)^{2}$$

In [17]:

factor_list(x**2*z + 4*x*y*z + 4*y**2*z)

Out[17]:

$$\begin{pmatrix}1, & \begin{bmatrix}\begin{pmatrix}z, & 1\end{pmatrix}, & \begin{pmatrix}x + 2 y, & 2\end{pmatrix}\end{bmatrix}\end{pmatrix}$$

In [18]:

expr = expand((cos(x) + sin(x))**2)

In [19]:

expr

Out[19]:

$$\sin^{2}{\left (x \right )} + 2 \sin{\left (x \right )} \cos{\left (x \right )} + \cos^{2}{\left (x \right )}$$

In [20]:

factor(expr)

Out[20]:

$$\left(\sin{\left (x \right )} + \cos{\left (x \right )}\right)^{2}$$

In [21]:

expr = x*y + x - 3 + 2*x**2 - z*x**2 + x**3

In [22]:

expr

Out[22]:

$$x^{3} - x^{2} z + 2 x^{2} + x y + x - 3$$

In [23]:

collected_expr = collect(expr, x)
collected_expr

Out[23]:

$$x^{3} + x^{2} \left(- z + 2\right) + x \left(y + 1\right) - 3$$

In [24]:

collected_expr.coeff(x,2)

Out[24]:

$$- z + 2$$

In [25]:

cancel((x**2 + 2*x + 1)/(x**2 + x))

Out[25]:

$$\frac{1}{x} \left(x + 1\right)$$

In [26]:

expr = 1/x + (3*x/2 - 2)/(x - 4)

In [27]:

expr

Out[27]:

$$\frac{\frac{3 x}{2} - 2}{x - 4} + \frac{1}{x}$$

In [28]:

cancel(expr)

Out[28]:

$$\frac{3 x^{2} - 2 x - 8}{2 x^{2} - 8 x}$$

In [29]:

expr = (x*y**2 - 2*x*y*z + x*z**2 + y**2 - 2*y*z + z**2)/(x**2 - 1)
expr

Out[29]:

$$\frac{1}{x^{2} - 1} \left(x y^{2} - 2 x y z + x z^{2} + y^{2} - 2 y z + z^{2}\right)$$

In [30]:

cancel(expr)

Out[30]:

$$\frac{1}{x - 1} \left(y^{2} - 2 y z + z^{2}\right)$$

In [31]:

factor(expr)

Out[31]:

$$\frac{\left(y - z\right)^{2}}{x - 1}$$

In [32]:

expr = (4*x**3 + 21*x**2 + 10*x + 12)/(x**4 + 5*x**3 + 5*x**2 + 4*x)
expr

Out[32]:

$$\frac{4 x^{3} + 21 x^{2} + 10 x + 12}{x^{4} + 5 x^{3} + 5 x^{2} + 4 x}$$

In [33]:

apart(expr)

Out[33]:

$$\frac{2 x - 1}{x^{2} + x + 1} - \frac{1}{x + 4} + \frac{3}{x}$$

In [34]:

cancel(Out[33])

Out[34]:

$$\frac{4 x^{3} + 21 x^{2} + 10 x + 12}{x^{4} + 5 x^{3} + 5 x^{2} + 4 x}$$

In [35]:

acos(x)

Out[35]:

$$\operatorname{acos}{\left (x \right )}$$

In [36]:

cos(acos(x))

Out[36]:

$$x$$

In [10]:

acos(cos(-5*pi/4))

Out[10]:

$$\frac{3 \pi}{4}$$

In [39]:

asin(1)

Out[39]:

$$\frac{\pi}{2}$$

In [40]:

trigsimp(sin(x)**2 + cos(x)**2)

Out[40]:

$$1$$

In [41]:

expand_trig(sin(x + y))

Out[41]:

$$\sin{\left (x \right )} \cos{\left (y \right )} + \sin{\left (y \right )} \cos{\left (x \right )}$$

In [42]:

expand_trig(cos(2*x))

Out[42]:

$$2 \cos^{2}{\left (x \right )} - 1$$

In [43]:

Out[42].subs(cos(x)**2,1-sin(x)**2)

Out[43]:

$$- 2 \sin^{2}{\left (x \right )} + 1$$

In [11]:

x, y = symbols('x y', positive=True)
a, b = symbols('a b', real=True)
z, t, c = symbols('z t c')

In [12]:

sqrt(x) == x**Rational(1, 2)

Out[12]:

True

In [13]:

powsimp(x**a*x**b)

Out[13]:

$$x^{a + b}$$

In [14]:

powsimp(x**a*y**a)

Out[14]:

$$\left(x y\right)^{a}$$

In [49]:

powsimp(t**c*z**c)

Out[49]:

$$t^{c} z^{c}$$

In [50]:

powsimp(t**c*z**c, force=True)

Out[50]:

$$\left(t z\right)^{c}$$

In [51]:

(z*t)**2

Out[51]:

$$t^{2} z^{2}$$

In [52]:

sqrt(x*y)

Out[52]:

$$\sqrt{x} \sqrt{y}$$

In [53]:

x, y, z = symbols('x y z')
k, m, n = symbols('k m n')

In [54]:

binomial(n, k)

Out[54]:

$${\binom{n}{k}}$$

In [55]:

hyper([1, 2], [3], z)

Out[55]:

$${{}_{2}F_{1}\left(\begin{matrix} 1, 2 \\ 3 \end{matrix}\middle| {z} \right)}$$

In [60]:

cosh(x).rewrite(exp)

Out[60]:

$$\frac{e^{x}}{2} + \frac{e^{- x}}{2}$$

In [61]:

acosh(x).rewrite(log)

Out[61]:

$$\operatorname{acosh}{\left (x \right )}$$

In [15]:

def list_to_frac(l):
    expr = Integer(0)
    for i in reversed(l[1:]):
        expr += i
        expr = 1/expr
    return l[0] + expr

In [16]:

list_to_frac([x,y,z])

Out[16]:

$$x + \frac{1}{y + \frac{1}{z}}$$

In [17]:

list_to_frac([1,2,3,4])

Out[17]:

$$\frac{43}{30}$$

In [18]:

list_to_frac([1,1,1,1,1,1,1]).evalf()

Out[18]:

$$1.61538461538462$$

In [72]:

l = [1,2,3,4,5]
l

Out[72]:

$$\begin{bmatrix}1, & 2, & 3, & 4, & 5\end{bmatrix}$$

In [79]:

Out[79]:

$$\begin{bmatrix}5, & 4, & 3, & 2, & 1\end{bmatrix}$$

In [81]:

l[1:]

Out[81]:

$$\begin{bmatrix}4, & 3, & 2, & 1\end{bmatrix}$$

In [126]:

x, y, z = symbols('x y z')

In [127]:

diff(cos(x),x)

Out[127]:

$$- \sin{\left (x \right )}$$

In [84]:

diff(exp(x**2),x)

Out[84]:

$$2 x e^{x^{2}}$$

In [94]:

diff(y*exp(x**2),x,y)

Out[94]:

$$2 x e^{x^{2}}$$

In [96]:

diff(sin(y)*exp(x**2),x,y,y,x,5)

Out[96]:

$$- 8 \left(8 x^{6} + 60 x^{4} + 90 x^{2} + 15\right) e^{x^{2}} \sin{\left (y \right )}$$

In [97]:

integrate(cos(x),x)

Out[97]:

$$\sin{\left (x \right )}$$

In [98]:

integrate(x*cos(x),)

Out[98]:

$$x \sin{\left (x \right )} + \cos{\left (x \right )}$$

In [19]:

integrate(x*cos(x**2),x)

Out[19]:

$$\frac{1}{2} \sin{\left (x^{2} \right )}$$

In [20]:

integrate(exp(-x), (x,0,oo))

Out[20]:

$$1$$

In [21]:

integrate(exp(-x**2 - y**2), (x,-oo,oo),(y,-oo,oo))

Out[21]:

$$\pi$$

In [106]:

expr = integrate(exp(exp(x)),x)

In [107]:

print expr

Integral(exp(exp(x)), x)

In [108]:

expr

Out[108]:

$$\int e^{e^{x}}\, dx$$

In [22]:

expr = Integral(log(x)**2,x)
expr

Out[22]:

$$\int \log^{2}{\left (x \right )}\, dx$$

In [23]:

expr.doit()

Out[23]:

$$x \log^{2}{\left (x \right )} - 2 x \log{\left (x \right )} + 2 x$$

In [111]:

integ = Integral(sin(x**2), x)
integ

Out[111]:

$$\int \sin{\left (x^{2} \right )}\, dx$$

In [112]:

integ.doit()

Out[112]:

$$\frac{3 \sqrt{2} \sqrt{\pi} S\left(\frac{\sqrt{2} x}{\sqrt{\pi}}\right)}{8 \Gamma{\left(\frac{7}{4} \right)}} \Gamma{\left(\frac{3}{4} \right)}$$

In [15]:

integ = Integral(x**y*exp(-x), (x, 0, oo))
integ

Out[15]:

$$\int_{0}^{\infty} x^{y} e^{- x}\, dx$$

In [16]:

integ.doit()

Out[16]:

$$\begin{cases} \Gamma{\left(y + 1 \right)} & \text{for}\: - \Re{y} < 1 \\\int_{0}^{\infty} x^{y} e^{- x}\, dx & \text{otherwise} \end{cases}$$

In [115]:

expr = x**2/exp(x)
expr.subs(x,oo)

Out[115]:

$$\mathrm{NaN}$$

In [116]:

limit(expr,x,oo)

Out[116]:

$$0$$

In [117]:

limit(1/x, x, 0, '+')

Out[117]:

$$\infty$$

In [125]:

limit(1/x, x, 0, '-')

Out[125]:

$$-\infty$$

In [24]:

r, theta, a, b = symbols('r, theta a b')

In [25]:

ans = integrate(sqrt(b**2-r**2)*r,(r,a,b),(theta,0,2*pi))

In [26]:

ans

Out[26]:

$$2 \pi \left(- \frac{a^{2}}{3} \sqrt{- a^{2} + b^{2}} + \frac{b^{2}}{3} \sqrt{- a^{2} + b^{2}}\right)$$

In [27]:

factor(ans)

Out[27]:

$$\frac{2 \pi}{3} \left(- \left(a - b\right) \left(a + b\right)\right)^{\frac{3}{2}}$$

In [11]:

x = symbols('x')

In [32]:

integrate(x*sqrt(x-1),x)

Out[32]:

$$\begin{cases} \frac{2 x^{2}}{5} \sqrt{x - 1} - \frac{2 x}{15} \sqrt{x - 1} - \frac{4}{15} \sqrt{x - 1} & \text{for}\: \left\lvert{x}\right\rvert > 1 \\\frac{2 i}{5} x^{2} \sqrt{- x + 1} - \frac{2 i}{15} x \sqrt{- x + 1} - \frac{4 i}{15} \sqrt{- x + 1} & \text{otherwise} \end{cases}$$

In [39]:

x = symbols('x')
myint = Integral(x*sqrt(x-1),(x,2,5))

In [28]:

(log(-4) + acos(3)).evalf(5)

Out[28]:

$$1.3863 + 4.9043 i$$

In [37]:

sqrt(2).evalf(30)

Out[37]:

$$1.41421356237309504880168872421$$

In [4]:

Rational(1,2) + Rational(1,3)

Out[4]:

$$\frac{5}{6}$$

In [6]:

expand((x+y)**5)

Out[6]:

$$x^{5} + 5 x^{4} y + 10 x^{3} y^{2} + 10 x^{2} y^{3} + 5 x y^{4} + y^{5}$$

In [9]:

trigsimp(sin(x)/cos(x))

Out[9]:

$$\tan{\left (x \right )}$$

In [10]:

limit(x**x,x,0)

Out[10]:

$$1$$

In [11]:

limit(sin(x)/x,x,0)

Out[11]:

$$1$$

In [12]:

diff(log(x),x)

Out[12]:

$$\frac{1}{x}$$

In [18]:

integrate(x**3, (x,-oo,0))

Out[18]:

$$-\infty$$

In [19]:

integrate(exp(-x**2),(x,-oo,oo))

Out[19]:

$$\sqrt{\pi}$$

In [36]:

factor(x**3 - x**2 + x + 1)

Out[36]:

$$x^{3} - x^{2} + x + 1$$

In [32]:

factor(x**2 + 1, extension=[I])

Out[32]:

$$\left(x - i\right) \left(x + i\right)$$

In [64]:

diff(y(x)**2-5*sin(x),x)

Out[64]:

$$2 y{\left (x \right )} \frac{d}{d x} y{\left (x \right )} - 5 \cos{\left (x \right )}$$

In [67]:

solve(Out[64],cos(x))

Out[67]:

[2*y(x)*Derivative(y(x), x)/5]

In [66]:

init_printing(pretty_print=False)

In [68]:

solve(Out[64],Derivative(y(x),x))

Out[68]:

[5*cos(x)/(2*y(x))]

In [69]:

yp = Out[68]

In [74]:

yp[0].subs([(y(x),3),(x,4)])

Out[74]:

5*cos(4)/6

In [2]:

integrate(x*y,(z,0,10-10*x-5*y/2),(y,0,4-4*x),(x,0,1))

Out[2]:

$$\frac{4}{3}$$

In [3]:

integrate(x*y,(z,0,10-10*x-5*y/2),(x,0,1-y/4),(y,0,4))

Out[3]:

$$\frac{4}{3}$$

In [13]:

solve(diff(12*x**5+60*x**4-100*x**3+2,x,2),x)

Out[13]:

$$\begin{bmatrix}0, & - \frac{3}{2} + \frac{\sqrt{19}}{2}, & - \frac{\sqrt{19}}{2} - \frac{3}{2}\end{bmatrix}$$

In [22]:

solns=solve(x**3 - 3*x**2 + 5*x - 7, x)

In [104]:

for soln in solve(x**3 - 3*x**2 + 5*x - 7, x):
    print soln.evalf(5),

0.41025 - 1.7445*I 0.41025 + 1.7445*I 2.1795

In [46]:

for soln in solns:
    print soln.evalf(5),

0.41025 - 1.7445*I 0.41025 + 1.7445*I 2.1795

In [105]:

diff(y**2*x**2,x)

Out[105]:

$$2 x \left(- \frac{g m}{k} t + \frac{m}{k^{2}} \left(1 - e^{- \frac{k t}{m}}\right) \left(g m + k v_{0} \sin{\left (th \right )}\right)\right)^{2}$$

In [48]:

m, k, g, v0, th, t= symbols('m k g v0 th t')
y = m/k*((g*m + v0*k*sin(th))/k)*(1 - exp(-k*t/m)) - m*g*t/k

In [50]:

yp=diff(y,t)

In [51]:

yp.subs()

Out[51]:

$$- \frac{g m}{k} + \frac{1}{k} \left(g m + k v_{0} \sin{\left (th \right )}\right) e^{- \frac{k t}{m}}$$

In [52]:

subs?

Object `subs` not found.

In [72]:

yps=yp.subs([(g,32),(k,1),(m,3),(v0,100),(th,pi/2)])
yps

Out[72]:

$$-96 + 196 e^{- \frac{t}{3}}$$

In [59]:

plot(yps,(t,0,5))

Out[59]:

<sympy.plotting.plot.Plot at 0x10680e5d0>

In [70]:

ys=y.subs([(g,32),(k,1),(m,3),(v0,100),(th,pi/2)])
ys

Out[70]:

$$- 96 t + 588 - 588 e^{- \frac{t}{3}}$$

In [71]:

plot(ys,(t,0,5))

Out[71]:

<sympy.plotting.plot.Plot at 0x106938650>

In [73]:

solve(yps,t)

Out[73]:

$$\begin{bmatrix}- 3 \log{\left (24 \right )} + 3 \log{\left (49 \right )}\end{bmatrix}$$

In [77]:

expand((2*x - 1)*(x + 1))

Out[77]:

$$2 x^{2} + x - 1$$

In [83]:

solve(ln(2*x) - ln(x-2),x)

Out[83]:

$$\begin{bmatrix}-2\end{bmatrix}$$

In [84]:

solve(ln(x) + ln(2*x+1), x)

Out[84]:

$$\begin{bmatrix}\frac{1}{2}\end{bmatrix}$$

In [86]:

(log(1/2)/5600).evalf(10)

Out[86]:

$$-0.0001237762822$$

In [97]:

soln=solve(x*exp(x) - 50, x)
soln[0].evalf(20)

Out[97]:

$$2.8608901779822108668$$

In [99]:

2.86089*exp(2.86089)

Out[99]:

$$49.9999879902819$$

In [110]:

solve(x*3**x-10,x)[0].evalf()

Out[110]:

$$1.6436099455207$$

In [111]:

plot(1/sqrt(cos(x)),(x,0,2*pi))

Out[111]:

<sympy.plotting.plot.Plot at 0x10b4dd190>

In [7]:

f = x*sqrt(x**2+1)
fp = simplify(diff(f,x))
fp

Out[7]:

$$\frac{2 x^{2} + 1}{\sqrt{x^{2} + 1}}$$

In [8]:

fpp = simplify(diff(fp,x))
fpp

Out[8]:

$$\frac{x \left(2 x^{2} + 3\right)}{\left(x^{2} + 1\right)^{\frac{3}{2}}}$$

In [10]:

soln=(5*e + 2)/(e - 1)

In [11]:

log(soln + 2) - log(soln - 5)

Out[11]:

$$1.0$$

In [13]:

x, y, t, r = symbols('x y t r')
int=integrate(5/sqrt(25-r**2),(r,0,sqrt(25/2)))

In [15]:

2*pi*int.evalf()

Out[15]:

$$7.85398163397448 \pi$$

In [16]:

solve(3*exp(3*x+4)-120,x)

Out[16]:

$$\begin{bmatrix}\log{\left (\frac{2 \sqrt[3]{5}}{e^{\frac{4}{3}}} \right )}, & \log{\left (- \frac{\sqrt[3]{5}}{e^{\frac{4}{3}}} - \frac{\sqrt{3} i}{e^{\frac{4}{3}}} \sqrt[3]{5} \right )}, & \log{\left (- \frac{\sqrt[3]{5}}{e^{\frac{4}{3}}} + \frac{\sqrt{3} i}{e^{\frac{4}{3}}} \sqrt[3]{5} \right )}\end{bmatrix}$$

In [30]:

ans = diff((1+x)/(1+exp(x)),x)

In [31]:

ans.subs(x,0)

Out[31]:

$$\frac{1}{4}$$

In [32]:

ans

Out[32]:

$$- \frac{\left(x + 1\right) e^{x}}{\left(e^{x} + 1\right)^{2}} + \frac{1}{e^{x} + 1}$$

In [33]:

a, b, t = symbols('a b t')
h = -a*exp(-t/3) - b*t + a
diff(h,t)

Out[33]:

$$\frac{a}{3} e^{- \frac{t}{3}} - b$$

In [35]:

Out[33].subs([(a,738),(b,96)])

Out[35]:

$$-96 + 246 e^{- \frac{t}{3}}$$

In [36]:

solve(Out[35],t)

Out[36]:

$$\begin{bmatrix}- 3 \log{\left (16 \right )} + 3 \log{\left (41 \right )}\end{bmatrix}$$

In [41]:

h = -738*exp(-t/3) - 96*t + 738
hp = diff(h,t)
soln=solve(hp,t)
height = h.subs(t,soln[0])
print soln[0], soln[0].evalf(), height, height.evalf()

-3*log(16) + 3*log(41) 2.82295003339358 -288*log(41) + 450 + 288*log(16) 178.996796794216

In [45]:

df=simplify(diff(x**3/(x**2-25),x))
ddf=simplify(diff(df,x))

In [46]:

ddf

Out[46]:

$$\frac{50 x \left(x^{2} + 75\right)}{x^{6} - 75 x^{4} + 1875 x^{2} - 15625}$$

In [2]:

limit(sqrt(x**2+6*x+2)-x,x,oo)

Out[2]:

$$3$$

In [4]:

limit(x**(7*sin(x)),x,0,'+')

Out[4]:

$$1$$

In [5]:

limit(x**4/(sin(x)-x),x,0)

Out[5]:

$$0$$

In [6]:

limit(sin(10*x)/tan(3*x),x,0)

Out[6]:

$$\frac{10}{3}$$

In [7]:

limit((10**x-3**x)/x,x,0)

Out[7]:

$$- \log{\left (3 \right )} + \log{\left (10 \right )}$$

In [8]:

limit((1-6/x)**x,x,oo)

Out[8]:

$$e^{-6}$$

In [9]:

limit((1-cos(6*x))/(1-cos(3*x)),x,0)

Out[9]:

$$4$$

In [11]:

limit(4**x - 3**x - 10*(x**2-1),x,1)

Out[11]:

$$1$$

In [12]:

limit(10*x*exp(1/x) - 10*x,x,oo)

Out[12]:

$$10$$

In [3]:

r, t = symbols('r t')
integrate(r**2 * cos(t)**2 * (1-r**2) * r, (r,0,1), (t,0,2*pi))

Out[3]:

$$\frac{\pi}{12}$$

In [8]:

x = symbols('x')
f = 2**(-x)*(x-10)**3
fp=diff(f,x)
factor(fp)
fpp=diff(fp,x)
factor(fpp)

Out[8]:

$$2^{- x} \left(x - 10\right) \left(x^{2} \log^{2}{\left (2 \right )} - 20 x \log^{2}{\left (2 \right )} - 6 x \log{\left (2 \right )} + 6 + 60 \log{\left (2 \right )} + 100 \log^{2}{\left (2 \right )}\right)$$

In [14]:

fpp_approx=Out[8].subs(log(2),log(2.0))

In [15]:

fpp_approx

Out[15]:

$$2^{- x} \left(x - 10\right) \left(0.480453013918201 x^{2} - 13.7679433617237 x + 95.6341322254169\right)$$

In [17]:

solve(0.48*x**2-13.77*x+95.63,x)

Out[17]:

$$\begin{bmatrix}11.7914966043057, & 16.8960033956944\end{bmatrix}$$

In [22]:

fp_crit=solve(fp,x)

In [24]:

fp_crit[1].evalf()

Out[24]:

$$14.3280851226669$$

In [26]:

factor(fp)

Out[26]:

$$- 2^{- x} \left(x - 10\right)^{2} \left(x \log{\left (2 \right )} - 10 \log{\left (2 \right )} - 3\right)$$

In [30]:

r, R = symbols('r R')
V = pi*R**2/3 * sqrt(r**2 - R**2)
dV = diff(V,R)
solve(dV,R)

Out[30]:

$$\begin{bmatrix}0, & - \frac{\sqrt{6} r}{3}, & \frac{\sqrt{6} r}{3}\end{bmatrix}$$

In [33]:

180/pi*2*pi*(1-sqrt(2/3.0)).evalf()

Out[33]:

$$66.0612308660186$$

In [35]:

simplify(V.subs(R,r*sqrt(6)/3))

Out[35]:

$$\frac{2 \pi}{27} \sqrt{3} r^{2} \sqrt{r^{2}}$$

In [14]:

from numpy import *

x = symbols('x')
f = x**3 + x - 2
fp = diff(f,x)

xi = [2]

ii = arange(0,10,1)

for i in {1,2,3,4,5,6}:
    xi.append(xi[-1] - f(xi[-1])/fp(xi[-1]))

---------------------------------------------------------------------------
TypeError                                 Traceback (most recent call last)
<ipython-input-14-c35f3b275e7f> in <module>()
      3 x = symbols('x')
      4 f = x**3 + x - 2
----> 5 fp = diff(f,x)
      6 
      7 xi = [2]

/Users/gary/.virtualenvs/scipy/lib/python2.7/site-packages/numpy/lib/function_base.pyc in diff(a, n, axis)
    982     if n == 0:
    983         return a
--> 984     if n < 0:
    985         raise ValueError(
    986                 "order must be non-negative but got " + repr(n))

/Users/gary/.virtualenvs/scipy/lib/python2.7/site-packages/sympy/core/relational.pyc in __nonzero__(self)
    109 
    110     def __nonzero__(self):
--> 111         raise TypeError("symbolic boolean expression has no truth value.")
    112 
    113     __bool__ = __nonzero__

TypeError: symbolic boolean expression has no truth value.

In [23]:

t = symbols('t')
P = 10000*exp(0.55*t)
dP = diff(P,t)
t0 = solve(dP - 34000, t)

print("P(5) = " + str(P.subs(t,5)))
print("P'(5) = " + str(dP.subs(t,5)))
print("t0 = " + str(t0[0]))
print("P(t0) = " + str(P.subs(t,t0[0])))

P(5) = 156426.318841882
P'(5) = 86034.4753630349
t0 = 3.31202260432316
P(t0) = 61818.1818181818

In [52]:

t, T, k = symbols('t T k', real=True)

T0 = 70
E = 325

T = (T0 - E)*exp(k*t) + E

ks = solve(T.subs(t,10)-100.0, k)[0]
print("ks = " + str(ks))

Ts = T.subs(k,-0.0125)
print(Ts)
print([(time,Ts.subs(t,time)) for time in {10, 20, 30, 40, 50, 60, 70, 80, 90, 100, 110, 120}])

ks = -0.0125163142954006
-255*exp(-0.0125*t) + 325
[(100, 251.941276800652), (70, 218.700184981980), (40, 170.334681773278), (10, 99.9632898409282), (110, 260.525903069790), (80, 231.190742501282), (50, 188.508335727657), (20, 126.405800316792), (120, 268.101809162150), (90, 242.213620823621), (60, 204.546529051041), (30, 149.741233908302)]

In [79]:

t, P, k = symbols('t P k', real=True)
ts = symbols('ts', real=True)

M = 1000.0
P0 = 8.0

P = M*P0*exp(k*t) / (M + P0*(exp(k*t) - 1))

dP = simplify(diff(P, t))

ks = solve(P.subs(t,12.0)-80.0,k)[0]
print(ks)

print(P.subs(k,ks))

print(P.subs([(t,12),(k,ks)]))
#plot(dP.subs(k,ks),(t,0,60))
#plot(P.subs(k,ks), (t,0,60))

print(dP)

#ts = solve(dP.subs(k,ks)-40, t)
#print(ts)

0.198161210852986
8000.0*exp(0.198161210852986*t)/(8.0*exp(0.198161210852986*t) + 992.0)
80.0000000000000
7936000.0*k*exp(k*t)/(8.0*exp(k*t) + 992.0)**2

In []:

Math Maniac's Musings

A Python Notebook with Sympy.

Sympy (Symbolic Python)