CoCalc -- demo notebook.ipynb

a simuw scratch worksheet

Project: Shared Project -- SIMUW 2019

Views: ³⁸⁵
License: APACHE

Kernel: SageMath (stable)

In [8]:

var('x y')
plot3d(sin(x) * cos(y), (x, -10, 10), (y, -10, 10))

Icosahedron

In [53]:

var('x y')
plot3d(tan(1/x) * sin(y^2) - cos(x), (x, -10, 10), (y, -5, 5))

In [50]:

plot(sin(1/x)*cos(x^2), (x,0,10))

In [2]:

show(icosahedron(color='green', opacity=.5, mesh=3), spin=1)

In [39]:

plot(cos(x)*x^2, (x,-15,15))

In [0]:

def f(x):
    print("x squared is", x^2)

f(10)

Icosahedron

In [4]:

show(icosahedron(color='green', opacity=.5, mesh=3), spin=1)

In [11]:

plot(x^3 * sin(x), (x,100,108))

Show a colored cube

In [0]:

thecube = cube(
  color=['red', 'blue', 'green'],
  frame_thickness=2,
  frame_color='brown',
  opacity=0.8
)
show(thecube, frame=False)

In [1]:

factor(2020)

2^2 * 5 * 101

Show a colored cube

In [0]:

thecube = cube(
  color=['red', 'blue', 'green'],
  frame_thickness=2,
  frame_color='brown',
  opacity=0.8
)
show(thecube, frame=False)

Torus

In [1]:

from sage.plot.plot3d.shapes import Torus
inner_radius = .3; outer_radius = 1
show(Torus(outer_radius, inner_radius, color='orange'), aspect_ratio=1, spin=3)

In [2]:

plot(sin(x^2)*cos(x), (x,0,10))

Symbolic expressions are also Python objects. This means, there are ways to inspect and take them apart.

In this example here, we start with an equation and "zoom in" to a small part of it.

Tipp: Combine this technique with substitutions to build new expressions out of existing ones.

Transitions and a plot of a Finite State Machine

In [40]:

fsm = Automaton([(0, 0, 0), (0, 1, 1), (0, 0, 2), (1, 2, 2), (2, 0, 0)], initial_states=[0, 1, 2], final_states=[0, 1])
for transition in fsm.accessible_components().transitions():
    print(transition)
fsm.plot()

Transition from 0 to 0: 0|-
Transition from 0 to 1: 1|-
Transition from 0 to 0: 2|-
Transition from 1 to 2: 2|-
Transition from 2 to 0: 0|-

Show a colored cube

In [0]:

thecube = cube(
  color=['red', 'blue', 'green'],
  frame_thickness=2,
  frame_color='brown',
  opacity=0.8
)
show(thecube, frame=False)

SageMath's preparser makes it easy to define functions.

In [15]:

from sage.plot.plot3d.shapes import Torus
inner_radius = .3; outer_radius = 1
show(Torus(outer_radius, inner_radius, color='blue'), aspect_ratio=1, spin=3)

In [6]:

plot(x^2)

In [39]:

plot(cos((1/2)*x)+sin(x), (x,0,10))

In [12]:

In [0]:

In [0]:

sage: from sage.symbolic.function_factory import deprecated_custom_evalf_wrapper as dcew
sage: def old_func(x, prec=0): print("x: %s, prec: %s" % (x, prec))
sage: new_func = dcew(old_func)
sage: new_func(5, parent=RR)
x: 5, prec: 53
sage: new_func(0r, parent=ComplexField(100))
x: 0, prec: 100

In [20]:

G = Cylinder(1, .5) + Cylinder(.25, 3).translate(0, 0, -3)

---------------------------------------------------------------------------
NameError                                 Traceback (most recent call last)
<ipython-input-20-5b4a735ee13e> in <module>()
----> 1 G = Cylinder(Integer(1), RealNumber('.5')) + Cylinder(RealNumber('.25'), Integer(3)).translate(Integer(0), Integer(0), -Integer(3))

NameError: name 'Cylinder' is not defined

In [19]:

sum(Cone(.1, sin(n)).translate(n, sin(n), 0) for n in [0..10, step=.1])

---------------------------------------------------------------------------
TypeError                                 Traceback (most recent call last)
<ipython-input-19-fc0dca25130a> in <module>()
----> 1 sum(Cone(RealNumber('.1'), sin(n)).translate(n, sin(n), Integer(0)) for n in (ellipsis_range(Integer(0),Ellipsis,Integer(10), step=RealNumber('.1'))))

/ext/sage/sage-8.7_1804/local/lib/python2.7/site-packages/sage/misc/functional.pyc in symbolic_sum(expression, *args, **kwds)
    575         return expression.sum(*args, **kwds)
    576     elif len(args) <= 1:
--> 577         return sum(expression, *args)
    578     else:
    579         from sage.symbolic.ring import SR
<ipython-input-19-fc0dca25130a> in <genexpr>((n,))
----> 1 sum(Cone(RealNumber('.1'), sin(n)).translate(n, sin(n), Integer(0)) for n in (ellipsis_range(Integer(0),Ellipsis,Integer(10), step=RealNumber('.1'))))

/ext/sage/sage-8.7_1804/local/lib/python2.7/site-packages/sage/geometry/cone.pyc in Cone(rays, lattice, check, normalize)
    427     # Cone from rays
    428     if check or normalize:
--> 429         rays = normalize_rays(rays, lattice)
    430     if lattice is None:
    431         if rays:
/ext/sage/sage-8.7_1804/local/lib/python2.7/site-packages/sage/geometry/cone.pyc in normalize_rays(rays, lattice)
    643         raise TypeError(
    644                     "rays must be given as a list or a compatible structure!"
--> 645                     "\nGot: %s" % rays)
    646     if rays:
    647         if lattice is None:
TypeError: rays must be given as a list or a compatible structure!
Got: 0.100000000000000

In [18]:

sum(Cone(.1, sin(n), color='yellow').translate(n, sin(n), 0) for n in [0..10, step=.1])

---------------------------------------------------------------------------
TypeError                                 Traceback (most recent call last)
<ipython-input-18-be3c25c908c1> in <module>()
----> 1 sum(Cone(RealNumber('.1'), sin(n), color='yellow').translate(n, sin(n), Integer(0)) for n in (ellipsis_range(Integer(0),Ellipsis,Integer(10), step=RealNumber('.1'))))

/ext/sage/sage-8.7_1804/local/lib/python2.7/site-packages/sage/misc/functional.pyc in symbolic_sum(expression, *args, **kwds)
    575         return expression.sum(*args, **kwds)
    576     elif len(args) <= 1:
--> 577         return sum(expression, *args)
    578     else:
    579         from sage.symbolic.ring import SR
<ipython-input-18-be3c25c908c1> in <genexpr>((n,))
----> 1 sum(Cone(RealNumber('.1'), sin(n), color='yellow').translate(n, sin(n), Integer(0)) for n in (ellipsis_range(Integer(0),Ellipsis,Integer(10), step=RealNumber('.1'))))

TypeError: Cone() got an unexpected keyword argument 'color'

In [0]:

var('x')
f(x) = 3 * x -1
print(f)
g(x) = (1 - x)^2
print(g)
# h is a composition of f and g
h = f(g)
print(h)
# evaluate h
print(h(x = 3))
print(h.subs(x = pi))

In [0]:

plot(1/x, (x,0,50))

Icosahedron

In [0]:

show(icosahedron(color='green', opacity=.5, mesh=3), spin=1)

In [0]:

plot(x^x, (x,0,10))

SageMath's preparser makes it easy to define functions.

In [47]:

plot(tan(x), (x,0,1000))

In [0]:

factor(1)

plot(sin(x)^(x^3),(x,-5,5))

In [0]:

factor(420420420)

In [12]:

factor(3)

3

In [3]:

plot(x, (x,0,10))

In [23]:

plot(2*x, (x,0,10))

In [0]:

In [0]:

In [26]:

plot(x, (x,0,10))

---------------------------------------------------------------------------
TypeError                                 Traceback (most recent call last)
<ipython-input-26-d1d15f9d8156> in <module>()
----> 1 plot()

/ext/sage/sage-8.7_1804/local/lib/python2.7/site-packages/sage/misc/decorators.pyc in wrapper(*args, **kwds)
    490                 options['__original_opts'] = kwds
    491             options.update(kwds)
--> 492             return func(*args, **options)
    493 
    494         #Add the options specified by @options to the signature of the wrapped
TypeError: plot() takes at least 1 argument (0 given)

In [0]:

plot(3*x, (0,10))

3D plot of $f(x,y) = \sin(x)\cos(y)$ .

In [3]:

var('x y')
plot3d(sin(x) * cos(y), (x, -10, 10), (y, -10, 10))

Show a colored cube

In [14]:

thecube = cube(
  color=['red', 'blue', 'green'],
  frame_thickness=2,
  frame_color='brown',
  opacity=0.8
)
show(thecube, frame=False)

In [40]:

plot (4^x, (0,10))

In [34]:

plot (x^(x*(1/50)),(x,0,10))

In [0]:

plot(()tan x),(x,0,30))

In [0]:

In [0]:

factor(2020)

Torus

In [13]:

from sage.plot.plot3d.shapes import Torus
inner_radius = .3; outer_radius = 1
show(Torus(outer_radius, inner_radius, color='orange'), aspect_ratio=1, spin=3)

In [0]: