501-Demonstration

︠2e8c85c1-e90f-4b7c-9e1b-780fbbce3603i︠
%html
<h1>Demonstration #2</h1>

<p>
You can decorate your worksheets with <emph>HTML</emph>.
</p>

<p>
Here is a link to the website of <a href="http://www.calvin.edu">a distinguished college</a>.  
</p>

<h2>Linear Algebra</h2>

︡78a31daa-a287-4947-b6eb-7beeaa127628︡{"html": "<h1>Demonstration #2</h1>\r\n\r\n<p>\r\nYou can decorate your worksheets with <emph>HTML</emph>.\r\n</p>\r\n\r\n<p>\r\nHere is a link to the website of <a href=\"http://www.calvin.edu\">a distinguished college</a>.  \r\n</p>\r\n\r\n<h2>Linear Algebra</h2>"}︡
︠1d11e61b-c44c-43a6-9631-e0139f447e29︠
A = matrix(3,3,[1,2,3,3,2,1,1,1,1])
show(A)
︡cf52079e-b474-4b6f-b2a8-258f7d3d4e80︡{"html": "<div class=\"math\">\\left(\\begin{array}{rrr}\n1 & 2 & 3 \\\\\n3 & 2 & 1 \\\\\n1 & 1 & 1\n\\end{array}\\right)</div>"}︡
︠24ae0e87-a7d2-423e-ad5e-89212ad19b2c︠
b = vector([0,-4,-1])
b
︡169cb145-aea2-40a2-9128-55d20866941a︡{"stdout": "(0, -4, -1)"}︡
︠f427faa6-86cc-4513-bba1-842aa25691a0︠
# Solve the equation Ax = b
#
A.solve_right(b)
︡f0662bee-7fa3-475f-8b5c-d2545373fcba︡{"html": "<span class=\"math\">\\left(-2,1,0\\right)</span>"}︡
︠a3d56101-a5c2-4b88-9043-8b1e1776a063︠
A.eigenvalues()
︡b0c3e229-0245-44f8-98f6-e22126b78405︡{"html": "<span class=\"math\">\\left[5, \n 0, \n -1\\right]</span>"}︡
︠84c83259-2226-4595-a55b-cad443b47b89︠
B = Matrix(2,2,[12,10,8,sqrt(7)])
B
︡ccda5e22-42d1-4f9b-9584-3ba13e5cb6b0︡{"html": "<span class=\"math\">\\left(\\begin{array}{rr}\n12 & 10 \\\\\n8 & \\sqrt{ 7 }\n\\end{array}\\right)</span>"}︡
︠2d8c8a80-7bf3-40a6-a2fb-4800b968f73c︠
B.det()
︡a5f5b67e-81b1-4f8d-a176-03e029fb192d︡{"html": "<span class=\"math\">{12 \\sqrt{ 7 }} - 80</span>"}︡
︠0c21c5dd-1c02-4d36-945f-aac0b56c2898︠
B.inverse()
︡57470c56-d6e2-4df3-9fe3-dfee813612a4︡{"html": "<span class=\"math\">\\left(\\begin{array}{rr}\n\\frac{\\sqrt{ 7 }}{{12 \\sqrt{ 7 }} - 80} & \\frac{-10}{{12 \\sqrt{ 7 }} - 80} \\\\\n\\frac{-8}{{12 \\sqrt{ 7 }} - 80} & \\frac{12}{{12 \\sqrt{ 7 }} - 80}\n\\end{array}\\right)</span>"}︡
︠370370cf-628c-424d-bd95-b10d9895a464︠
B^2
︡8e4de742-afab-4bb2-8314-d2a8bc5bb8df︡{"html": "<span class=\"math\">\\left(\\begin{array}{rr}\n224 & {10 \\sqrt{ 7 }} + 120 \\\\\n{8 \\sqrt{ 7 }} + 96 & 87\n\\end{array}\\right)</span>"}︡
︠473297c3-2ab7-4b0f-b885-9e2b87308e14i︠
%html
<h2>Abstract Algebra</h2>

︡b18e17e8-5eac-4587-aa5e-7a29e82d45ad︡{"html": "<h2>Abstract Algebra</h2>"}︡
︠08bb0fe8-cf42-4929-99a0-e2571298c4aa︠
G = AlternatingGroup(4)
G.gens()
︡46a226f1-ac27-40f1-ae97-e3628d63cc52︡{"html": "<span class=\"math\">\\left[(1,2,3), \n (2,3,4)\\right]</span>"}︡
︠9d76a71d-7211-432b-8a22-f6843fb2c2ff︠
G.cayley_graph().show3d()
︡e282b987-0455-4f31-837a-8b82ac2f0bdc︡︡
︠9a944203-86bb-4fca-96d1-139837937840︠
H = DihedralGroup(6)
H
︡753c9e56-9d1e-4bab-baa0-51d5c57a0297︡{"html": "<span class=\"math\">\\langle (1,2,3,4,5,6), (1,6)(2,5)(3,4) \\rangle</span>"}︡
︠9f594abd-fcdb-4504-8be0-e5b90d0d6df5︠
H.cayley_table()
︡7f55960b-abf8-49d0-a7cd-98a6ca9c0cd9︡{"html": "<span class=\"math\">\\left(\\begin{array}{rrrrrrrrrrrr}\nx_{0} & x_{1} & x_{2} & x_{3} & x_{4} & x_{5} & x_{6} & x_{7} & x_{8} & x_{9} & x_{10} & x_{11} \\\\\nx_{1} & x_{0} & x_{3} & x_{2} & x_{5} & x_{4} & x_{7} & x_{6} & x_{9} & x_{8} & x_{11} & x_{10} \\\\\nx_{2} & x_{10} & x_{0} & x_{4} & x_{3} & x_{6} & x_{5} & x_{8} & x_{7} & x_{11} & x_{1} & x_{9} \\\\\nx_{3} & x_{11} & x_{1} & x_{5} & x_{2} & x_{7} & x_{4} & x_{9} & x_{6} & x_{10} & x_{0} & x_{8} \\\\\nx_{4} & x_{9} & x_{10} & x_{6} & x_{0} & x_{8} & x_{3} & x_{11} & x_{5} & x_{1} & x_{2} & x_{7} \\\\\nx_{5} & x_{8} & x_{11} & x_{7} & x_{1} & x_{9} & x_{2} & x_{10} & x_{4} & x_{0} & x_{3} & x_{6} \\\\\nx_{6} & x_{7} & x_{9} & x_{8} & x_{10} & x_{11} & x_{0} & x_{1} & x_{3} & x_{2} & x_{4} & x_{5} \\\\\nx_{7} & x_{6} & x_{8} & x_{9} & x_{11} & x_{10} & x_{1} & x_{0} & x_{2} & x_{3} & x_{5} & x_{4} \\\\\nx_{8} & x_{5} & x_{7} & x_{11} & x_{9} & x_{1} & x_{10} & x_{2} & x_{0} & x_{4} & x_{6} & x_{3} \\\\\nx_{9} & x_{4} & x_{6} & x_{10} & x_{8} & x_{0} & x_{11} & x_{3} & x_{1} & x_{5} & x_{7} & x_{2} \\\\\nx_{10} & x_{2} & x_{4} & x_{0} & x_{6} & x_{3} & x_{8} & x_{5} & x_{11} & x_{7} & x_{9} & x_{1} \\\\\nx_{11} & x_{3} & x_{5} & x_{1} & x_{7} & x_{2} & x_{9} & x_{4} & x_{10} & x_{6} & x_{8} & x_{0}\n\\end{array}\\right)</span>"}︡
︠ba4b4abb-6217-490f-9b49-5896efd4a497︠
H.is_abelian()
︡9df57df4-f162-46df-812e-42dd29d28a57︡{"html": "<span class=\"math\">{\\rm False}</span>"}︡
︠5b3fb004-1f42-4bff-9225-4aed2081ffc5i︠
%html
<h2>Interactive Graphs</h2>

︡b0679c79-dee2-49b8-8911-30b281a77d9d︡{"html": "<h2>Interactive Graphs</h2>"}︡
︠90709306-f451-4029-9ded-01713dc07a56︠
@interact
def _(m=(-5..5), b=(-5..5)):
   show(plot((m*x+b), (-10,10)))
︡617a9e96-6f1e-49d8-b1e8-dd87369e1782︡{"html": "<!--notruncate--><div padding=6 id='div-interact-39'> <table width=800px height=20px bgcolor='#c5c5c5'\n                 cellpadding=15><tr><td bgcolor='#f9f9f9' valign=top align=left><table><tr><td align=right><font color=\"black\">m&nbsp;</font></td><td><table><tr><td>\n    \t<div id='slider-m-39' class='ui-slider ui-slider-3' style='margin:0px;'><span class='ui-slider-handle'></span></div>\n    \t</td><td><font color='black' id='slider-m-39-lbl'></font></td></tr></table><script>(function(){ var values = [\"-5\",\"-4\",\"-3\",\"-2\",\"-1\",\"0\",\"1\",\"2\",\"3\",\"4\",\"5\"]; setTimeout(function() {\n    $('#slider-m-39').slider({\n    \tstepping: 1, min: 0, max: 10, startValue: 0,\n    \tchange: function (e,ui) { var position = ui.value; if(values!=null) $('#slider-m-39-lbl').text(values[position]); interact(39, \"sage.server.notebook.interact.update(39, \\\"m\\\", 3, sage.server.notebook.interact.standard_b64decode(\\\"\"+encode64(position)+\"\\\"), globals());sage.server.notebook.interact.recompute(39)\"); },\n    \tslide: function(e,ui) { if(values!=null) $('#slider-m-39-lbl').text(values[ui.value]); }\n    });\n    if(values != null) $('#slider-m-39-lbl').text(values[$('#slider-m-39').slider('value')]);\n    }, 1); })();</script></td></tr>\n<tr><td align=right><font color=\"black\">b&nbsp;</font></td><td><table><tr><td>\n    \t<div id='slider-b-39' class='ui-slider ui-slider-3' style='margin:0px;'><span class='ui-slider-handle'></span></div>\n    \t</td><td><font color='black' id='slider-b-39-lbl'></font></td></tr></table><script>(function(){ var values = [\"-5\",\"-4\",\"-3\",\"-2\",\"-1\",\"0\",\"1\",\"2\",\"3\",\"4\",\"5\"]; setTimeout(function() {\n    $('#slider-b-39').slider({\n    \tstepping: 1, min: 0, max: 10, startValue: 0,\n    \tchange: function (e,ui) { var position = ui.value; if(values!=null) $('#slider-b-39-lbl').text(values[position]); interact(39, \"sage.server.notebook.interact.update(39, \\\"b\\\", 4, sage.server.notebook.interact.standard_b64decode(\\\"\"+encode64(position)+\"\\\"), globals());sage.server.notebook.interact.recompute(39)\"); },\n    \tslide: function(e,ui) { if(values!=null) $('#slider-b-39-lbl').text(values[ui.value]); }\n    });\n    if(values != null) $('#slider-b-39-lbl').text(values[$('#slider-b-39').slider('value')]);\n    }, 1); })();</script></td></tr>\n</table><div id='cell-interact-39'><?__SAGE__START>\n        <table border=0 bgcolor='#white' width=100% height=100%>\n        <tr><td bgcolor=white align=left valign=top><pre><?__SAGE__TEXT></pre></td></tr>\n        <tr><td  align=left valign=top><?__SAGE__HTML></td></tr>\n        </table><?__SAGE__END></div></td>\n                 </tr></table></div>\n                 "}︡
︠c9b83fe9-e86b-4328-b7d7-8a473586cd57︠
var('x')
x0    = 0
f(x)  = sin(x)*e^(-x)
p     = plot(f,-1,5, thickness=2)
dot   = point((x0,f(x0)),pointsize=80,rgbcolor=(1,0,0))

@interact
def _(order=(1..12)):
  ft = f.taylor(x,x0,order)
  pt = plot(ft,-1, 5, color='green', thickness=2)
  html('$f(x)\;=\;%s$'%latex(f))
  html('$\hat{f}(x;%s)\;=\;%s+\mathcal{O}(x^{%s})$'%(x0,latex(ft),order+1))
  show(dot + p + pt, ymin = -.5, ymax = 1)
︡89e3aa5c-b95d-4003-82b5-9cf265dcda85︡{"html": "<!--notruncate--><div padding=6 id='div-interact-11'> <table width=800px height=20px bgcolor='#c5c5c5'\n                 cellpadding=15><tr><td bgcolor='#f9f9f9' valign=top align=left><table><tr><td align=right><font color=\"black\">order&nbsp;</font></td><td><table><tr><td>\n    \t<div id='slider-order-11' class='ui-slider ui-slider-3' style='margin:0px;'><span class='ui-slider-handle'></span></div>\n    \t</td><td><font color='black' id='slider-order-11-lbl'></font></td></tr></table><script>(function(){ var values = [\"1\",\"2\",\"3\",\"4\",\"5\",\"6\",\"7\",\"8\",\"9\",\"10\",\"11\",\"12\"]; setTimeout(function() {\n    $('#slider-order-11').slider({\n    \tstepping: 1, min: 0, max: 11, startValue: 0,\n    \tchange: function (e,ui) { var position = ui.value; if(values!=null) $('#slider-order-11-lbl').text(values[position]); interact(11, \"sage.server.notebook.interact.update(11, \\\"order\\\", 5, sage.server.notebook.interact.standard_b64decode(\\\"\"+encode64(position)+\"\\\"), globals());sage.server.notebook.interact.recompute(11)\"); },\n    \tslide: function(e,ui) { if(values!=null) $('#slider-order-11-lbl').text(values[ui.value]); }\n    });\n    if(values != null) $('#slider-order-11-lbl').text(values[$('#slider-order-11').slider('value')]);\n    }, 1); })();</script></td></tr>\n</table><div id='cell-interact-11'><?__SAGE__START>\n        <table border=0 bgcolor='#white' width=100% height=100%>\n        <tr><td bgcolor=white align=left valign=top><pre><?__SAGE__TEXT></pre></td></tr>\n        <tr><td  align=left valign=top><?__SAGE__HTML></td></tr>\n        </table><?__SAGE__END></div></td>\n                 </tr></table></div>\n                 "}︡
︠ba714739-12bb-4b96-897e-4b7c7786b671i︠
%html
<h2>Contour Plots</h2>

︡b0ad722e-437e-4a5f-a50c-618a02a05610︡{"html": "<h2>Contour Plots</h2>"}︡
︠d76d970e-56d4-4d9e-a3c5-d518d6da6f34︠
x,y = var('x,y')
contour_plot(cos(x^2+y^2), (x,-4, 4), (y,-4, 4))
︡9d925800-dd7c-418b-9698-e73e8d7f2b6d︡︡
︠ea5d0fcb-3de9-4210-8f88-0fed7fa427ca︠
contour_plot(lambda x,y: cos(x*y), (-4, 4), (-4, 4))
︡e0c3123c-cbc4-4e71-8ebc-72a006e1762e︡︡
︠34816c62-3169-4c46-99ea-bc29f7cb5ff8︠
contour_plot(y^2 + 1 - x^3 - x + 7*sin(y), (x,-pi,pi), (y,-pi,pi))
︡5becf965-588f-4c5c-a759-ce787bf46c00︡︡
︠46f867e7-2a97-4665-be97-d51ca9d052d4i︠
%html
<h2>Parametric Plots</h2>

︡7d0fd6f3-6573-47a2-8846-5a08a0678af4︡{"html": "<h2>Parametric Plots</h2>"}︡
︠dad0386a-5060-4cf7-b68f-d3b8a16ecc18︠
t = var('t')
parametric_plot((3*sin(2*t), cos(t)),0,3*pi/2)
︡5d5885bc-d063-4b4a-b943-b2f1ebd51f9f︡︡
︠b3f5869a-9816-40d4-a227-775ca9fbef24︠
parametric_plot([cos(x) + 2 * cos(x/4), sin(x) - 2 * sin(x/4)], 0, 8*pi, fill = True)
︡7e440c61-136b-41e0-a45c-137d7ce50036︡︡
︠b5d42cf8-5f0c-4753-96bb-a38ac4efec4d︠
parametric_plot( (5*cos(x), 5*sin(x), x), (x,-12, 12), plot_points=150, color="red")
︡adabd4cc-7173-4c3a-b884-492d0d047238︡︡
︠9b401478-6879-47a2-b1ef-497dc25fca20i︠
%html
<h2>Platonic Solids</h2>

︡e1d27ffb-3bea-472e-b630-fc30dcc86aca︡{"html": "<h2>Platonic Solids</h2>"}︡
︠a3393680-641e-4594-9988-9d471c0e01e7︠
G = tetrahedron((0,-3.5,0), color='blue') + cube((0,-2,0),color=(.25,0,.5)) +\
      octahedron(color='red') + dodecahedron((0,2,0), color='orange') +\
      icosahedron(center=(0,4,0), color='yellow')

G.show(aspect_ratio=[1,1,1])
︡7b96a4cf-ab43-4fad-ba41-0fc096a691e5︡︡
︠9ffbcf4c-9da8-4e7b-b5e2-e58b36a22461i︠
%html
<h2>Implementing Algorithms</h2>

︡0073581e-9dba-499f-9760-4d0026738237︡{"html": "<h2>Implementing Algorithms</h2>"}︡
︠9c9197ea-796a-462c-acc6-b19b1bc693a1︠
def rungeKutta4(f, t0, tlast, y0, N, approx=False):
  """Approximately solve an IVP using the 4th-order Runge-Kutta method."""
  h = (tlast-t0)/N
  y = [y0]
  t = [t0 + j*h for j in [0..N]]
  for j in [0..N-1]:
    F1 = f(t[j], y[j])
    F2 = f(t[j] + h/2, y[j] + F1*h/2)
    F3 = f(t[j] + h/2, y[j] + F2*h/2)
    F4 = f(t[j+1], y[j] + h*F3)
    y.append(y[j] + h/6*(F1 + 2*(F2 + F3) + F4))
    if approx: y[j+1] = numerical_approx(y[j+1])
  return y
︡524da04b-f68b-4f42-95f3-681984523887︡︡
︠a19dc6a1-973e-418e-b438-31c64a56c619︠
y = rungeKutta4(lambda t,y: (t*y - y^2)/t^2, 1, 2, 2, 10, approx=True)
t = srange(1, 21/10, 1/10)

P = point(zip(t,y), rgbcolor='red', pointsize=32) + \
    plot(lambda t: 2*t/(1+2*ln(t)), 1, 2)

P.show()
︡b9dc020b-b289-46e9-a32f-cb5c9d2c18b1︡︡
︠283cb894-0784-4bc5-9b32-adcf88ff2e33i︠
%html
<h2>Call other languages from within Sage</h2>

︡ba8fcd85-db5a-4a7e-bb6d-b02ab0eea6a6︡{"html": "<h2>Call other languages from within Sage</h2>"}︡
︠84145ed5-662d-4407-8ce1-059ec6a2da37︠
%sh

echo You can run commands in languages other than Python!
echo ---
︡27392996-3414-46f4-9002-b5d8301d938e︡{"stdout": "You can run commands in languages other than Python!\n---\n/home/sage/sagenb/sage_notebook/worksheets/gregc/13/cells/1"}︡
︠85cbd972-9509-413f-b814-eaaf737a74c5︠
octave("[1 2 3]'")
︡0161f23c-4d59-48e2-8d67-e0f21141d3c5︡︡
︠17030474-a377-41d2-98e9-a275df71a797︠
f(x) = x^3 + 2*sqrt(x) + e^(i*x)
︡b5fd71d0-b5fd-4458-914d-ad27933b815a︡︡
︠eb8746c8-15c7-4a74-b995-f9035c39a0a7︠
latex(f(x))
︡9cb3b8c2-29a3-403e-90cb-414fa25f96ef︡{"stdout": "{e}^{{i x}}  + {x}^{3}  + {2 \\sqrt{ x }}"}︡
︠e3277b92-379b-4621-8352-afc0d1db044f︠
mathematica(f(x))
︡f8f43b6d-6e2e-446f-b3ca-cc4a572f6b44︡︡
︠7793d0e4-fc52-451c-b00c-3f8c855353f7︠
%mathematica
E^(I*x) + 2*Sqrt[x] + x^3
︡cb329828-a4e8-4a37-92a2-d5893728793f︡︡
︠501829c9-7849-4d68-96cd-fb18bdd1057c︠
# And you can import any python package:

from numpy import arange

arange(0, 10, 7/10)
︡59a71ba4-a6cd-481e-bfcb-ec0d5ece1c98︡{"stdout": "array([0, 7/10, 7/5, 21/10, 14/5, 7/2, 21/5, 49/10, 28/5, 63/10, 7, 77/10,\n       42/5, 91/10, 49/5], dtype=object)"}︡
︠b5831e44-6555-4130-ae93-8de2e824b231i︠
%html
<h2>Sage is continually getting better!</h2>

︡993141db-613b-4111-bbc2-9e18c33f3eee︡{"html": "<h2>Sage is continually getting better!</h2>"}︡
︠ac53b77d-07ee-45f9-8a69-ecf67b020dfe︠
Sage().version()
︡c4bc3776-4bbb-4883-b1e1-e650d167dba9︡{"html": "<span class=\"math\">\\text{Sage Version 3.4.1, Release Date: 2009-04-21}</span>"}︡