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%md

<H1>This is a sage worksheet.</H1>
 

<font color = purple>I have added this text by using the "markdown" indicator at the start of this command. Sage knows html and latex, so I can adopt a literate programming style, mixing text, equations like $y = \frac{x}{x-1}$, and commands -- as you can with for-purchase Computer Math Systems.  To see what I really typed here, double click on this text! To re-execute, highlight the command and press shift-return.</font>
︡912010a3-cb1a-43d7-9f82-81c84950015a︡{"md":"\n<H1>This is a sage worksheet.</H1>\n \n\n<font color = purple>I have added this text by using the \"markdown\" indicator at the start of this command. Sage knows html and latex, so I can adopt a literate programming style, mixing text, equations like $y = \\frac{x}{x-1}$, and commands -- as you can with for-purchase Computer Math Systems.  To see what I really typed here, double click on this text! To re-execute, highlight the command and press shift-return.</font>\n"}︡
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%md
<font color = green>Below are some commands that illustrate the similarities to Maple/Mathematica.  You simply click on the command and then shift-return to execute the command.

Notice also that once you execute a command you are moved to the next one -- or if you execute the last command, you will be moved to a blank line.
︡a40b0352-77a6-4712-9a08-6999ab4bf2c8︡{"md":"<font color = green>Below are some commands that illustrate the similarities to Maple/Mathematica.  You simply click on the command and then shift-return to execute the command.\n\nNotice also that once you execute a command you are moved to the next one -- or if you execute the last command, you will be moved to a blank line.\n"}︡
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5 + 6 - sin(4).n()
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5+6-sin(4.0)  #show n() and .n()  sin() is an object and you can attach a method to it... sin(4).n()
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f(x)=x^2 - 3*x + 2 # show f.<tab>
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show(f)
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f.abs
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limit(f(x), x=2)
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diff(f(x), x)
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integrate(f(x), x)
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plot(f(x), -10, 10, ymin=-2, ymax=2) #show ymin and ymax
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find_root(f(x), -10, 10)
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factor(f(x))
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solve(f(x)==0, x)
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var('v')
solve(v^2 - 2*v +2 == 0, v) # show how to add a new executable and do this one...show var('v') and del(v)
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%md
<font color = green>Sage has matlab-like functionality as well...</font>
︡12442600-93f1-4d7a-97b3-e94138a9a41f︡{"md":"<font color = green>Sage has matlab-like functionality as well...</font>\n"}︡
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u = vector([1, 3/2, 2, 4])
v = vector([-2, 1, 2, 3])  #show u. <tab>
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2*u - 3*v
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u[0]
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u.dot_product(v)
u.norm()
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A = matrix([[1, 2, 3], [4, 5, 6], [9, 8, 7]])
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show(A)
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A. transpose()
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A.echelon_form()
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A^-1
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%md
<font color = magenta>
The Python Programming Environment is available as well.  Here is a simple version of Newton's Method built as a function to call...
</font>
︡88cd6b40-e101-4aaa-bc77-c3e21fe250c2︡{"md":"<font color = magenta>\nThe Python Programming Environment is available as well.  Here is a simple version of Newton's Method built as a function to call...\n</font>\n"}︡
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def Newton(f, x0):
    g(x) = x - f(x) / diff (f(x), x)  #this is cheating, since I used the symbolic derivative...
    k = 0
    xold = x0
    xnew = g(x0)
    print k, xold
    while abs(xold - xnew) > 10E-12 and k < 10:
        xold = xnew
        xnew = g(xold)
        k += 1
        print k, xnew
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%md
Now let's call the function:
︡deb95420-79eb-4b35-a053-9cf702673406︡{"md":"Now let's call the function:\n"}︡
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w(x) = cos(x) - x^2
Newton(lambda x: w, 2.0)  #demonstrate show!
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%md
You can also access R (another freely available environment used mainly by statisticians) from within Sage; essentially you are passing information to the r interpreter...
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%md
Also built in to Sage is the programming language R
︡f839b362-d3fa-4f94-8670-d3870b583f50︡{"md":"Also built in to Sage is the programming language R\n"}︡
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r.summary(r.c(1,2,3,111,2,3,2,3,2,5,4))
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%md

More plotting...
︡fb651518-df3d-4089-93cc-188ec3512643︡{"md":"\nMore plotting...\n"}︡
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#You can also combine plots by adding them. This allows us to make each plot a different color.
plot(x^2, (x, 0, pi),color='red')+plot(sqrt(x), (x, 0, pi),color='green')+plot(sin(x), (x, 0, pi))
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plot3d?
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sombrero(x,y) = cos(sqrt(x^2 + y^2))
plot3d(sombrero, (x, -5, 5), (y, -5, 5))
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z = sqrt(-1)
print z, z^2, z^3, z^4
print z + 2*z^2 + 3*z^3 + 4*z^4
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%md
<h1>
http://sagemath.org/
</br>
http://wiki.sagemath.org/quickref
</H1>
︡ff748b31-d55f-431b-b3d4-42c61768c87c︡{"md":"<h1>\nhttp://sagemath.org/\n</br>\nhttp://wiki.sagemath.org/quickref\n</H1>\n"}︡
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