Differences between revisions 40 and 116 (spanning 76 versions)
⇤ ← Revision 40 as of 2011-06-17 01:43:57 →
Size: 56871
Editor: JohnTravis
Comment:
← Revision 116 as of 2020-08-11 14:08:17 → ⇥
Size: 63129
Editor: kcrisman
Comment:
Deletions are marked like this. Additions are marked like this.
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{{{ {{{#!sagecell
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            raise ValueError, "f must have a sign change in the interval (%s,%s)"%(a,b)             raise ValueError("f must have a sign change in the interval (%s,%s)"%(a,b))
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html("<h1>Double Precision Root Finding Using Bisection</h1>")
@interact
def _(f = cos(x) - x, a = float(0), b = float(1), eps=(-3,(-16..-1))):		 pretty_print(html("<h1>Double Precision Root Finding Using Bisection</h1>"))
@interact
def _(f = cos(x) - x, a = float(0), b = float(1), eps=(-3,(-16, -1))):
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     print "eps = %s"%float(eps)      print("eps = %s" % float(eps))
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         time c, intervals = bisect_method(f, a, b, eps)          c, intervals = bisect_method(f, a, b, eps)
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         print "f must have opposite sign at the endpoints of the interval"          print("f must have opposite sign at the endpoints of the interval")
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         print "root =", c
         print "f(c) = %r"%f(c)
         print "iterations =", len(intervals)		          print("root =", c)
         print("f(c) = %r" % f(x=c))
         print("iterations =", len(intervals))
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http://sagenb.org/home/pub/2824/

{{{		 https://cloud.sagemath.com/projects/19575ea0-317e-402b-be57-368d04c113db/files/pub/2801-2901/2824-Double%20Precision%20Root%20Finding%20Using%20Newton's%20Method.sagews

{{{#!sagecell
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    for i in xrange(maxiter):     for i in range(maxiter):
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html("<h1>Double Precision Root Finding Using Newton's Method</h1>")
@interact
def _(f = x^2 - 2, c = float(0.5), eps=(-3,(-16..-1)), interval=float(0.5)):		 pretty_print(html("<h1>Double Precision Root Finding Using Newton's Method</h1>"))
@interact
def _(f = x^2 - 2, c = float(0.5), eps=(-3,(-16, -1)), interval=float(0.5)):
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     print "eps = %s"%float(eps)
     time z, iterates = newton_method(f, c, eps)
     print "root =", z
     print "f(c) = %r"%f(x=z)		      print("eps = %s"%float(eps))
     z, iterates = newton_method(f, c, eps)
     print("root = {}".format(z))
     print("f(c) = %r" % f(x=z))
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     print "iterations =", n
     html(iterates)		      print("iterations = {}".format(n))
     pretty_print(html(iterates))
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http://sagenb.org/home/pub/2823/

{{{		 https://cloud.sagemath.com/projects/19575ea0-317e-402b-be57-368d04c113db/files/pub/2801-2901/2823.sagews

{{{#!sagecell
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{{{
html('<h2>Tangent line grapher</h2>')		 {{{#!sagecell
pretty_print(html('<h2>Tangent line grapher</h2>'))
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    tanf = f(x0i) + df(x0i)*(x-x0i)     tanf = f(x=x0i) + df(x=x0i)*(x-x0i)
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    print 'Tangent line is y = ' + tanf._repr_()     print('Tangent line is y = ' + tanf._repr_())
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    fmax = f.find_maximum_on_interval(prange[0], prange[1])[0]
    fmin = f.find_minimum_on_interval(prange[0], prange[1])[0]		     fmax = f.find_local_maximum(prange[0], prange[1])[0]
    fmin = f.find_local_minimum(prange[0], prange[1])[0]
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{{{ {{{#!sagecell
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    midys = [func(x_val) for x_val in midxs]     midys = [func(x=x_val) for x_val in midxs]
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    min_y = find_minimum_on_interval(func,a,b)[0]
    max_y = find_maximum_on_interval(func,a,b)[0]
    html('<h3>Numerical integrals with the midpoint rule</h3>')
    html('$\int_{a}^{b}{f(x) dx} {\\approx} \sum_i{f(x_i) \Delta x}$')
    print "\n\nSage numerical answer: " + str(integral_numerical(func,a,b,max_points = 200)[0])
    print "Midpoint estimated answer: " + str(RDF(dx*sum([midys[q] for q in range(n)])))		     min_y = min(0, find_local_minimum(func,a,b)[0])
    max_y = max(0, find_local_maximum(func,a,b)[0])
    pretty_print(html('<h3>Numerical integrals with the midpoint rule</h3>'))
    pretty_print(html(r'$\int_{a}^{b}{f(x) dx} {\approx} \sum_i{f(x_i) \Delta x}$'))
    print("\n\nSage numerical answer: " + str(integral_numerical(func,a,b,max_points = 200)[0]))
    print("Midpoint estimated answer: " + str(RDF(dx*sum([midys[q] for q in range(n)]))))
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{{{
# by Nick Alexander (based on the work of Marshall Hampton)
 {{{#!sagecell
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    t = sage.calculus.calculus.var('t')     t = var('t')
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            x = find_maximum_on_interval(func, q*dx + a, q*dx + dx + a)[1]             x = find_local_maximum(func, q*dx + a, q*dx + dx + a)[1]
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            x = find_minimum_on_interval(func, q*dx + a, q*dx + dx + a)[1]             x = find_local_minimum(func, q*dx + a, q*dx + dx + a)[1]
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    min_y = min(0, find_minimum_on_interval(func,a,b)[0])
    max_y = max(0, find_maximum_on_interval(func,a,b)[0])

    # html('<h3>Numerical integrals with the midpoint rule</h3>')		     min_y = min(0, find_local_minimum(func,a,b)[0])
    max_y = max(0, find_local_maximum(func,a,b)[0])

    pretty_print(html('<h3>Numerical integrals with the midpoint rule</h3>'))
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    sum_html = "%s \cdot \\left[ %s \\right]" % (dx, ' + '.join([ "f(%s)" % cap(i) for i in xs ]))
    num_html = "%s \cdot \\left[ %s \\right]" % (dx, ' + '.join([ str(cap(i)) for i in ys ]))		     sum_html = "%s \\cdot \\left[ %s \\right]" % (dx, ' + '.join([ "f(%s)" % cap(i) for i in xs ]))
    num_html = "%s \\cdot \\left[ %s \\right]" % (dx, ' + '.join([ str(cap(i)) for i in ys ]))
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    html(r'''
    <div class="math">
    \begin{align*}
   \int_{a}^{b} {f(x) \, dx} & = %s \\\
   \sum_{i=1}^{%s} {f(x_i) \, \Delta x}
      & = %s \\\
   & = %s \\\
   & = %s .
    \end{align*}
    </div>
    ''' % (numerical_answer, number_of_subdivisions, sum_html, num_html, estimated_answer))		     pretty_print(html(r'''
    <div class="math"> 
    \begin{align*}   \int_{a}^{b} {f(x) \, dx} & = %s \\\   \sum_{i=1}^{%s} {f(x_i) \, \Delta x} & = %s \\\   & = %s \\\   & = %s . \end{align*} </div>'''
                      % (numerical_answer, number_of_subdivisions, sum_html, num_html, estimated_answer)))
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{{{ {{{#!sagecell
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    html('$r=' + latex(b+sin(a1*t)^n1 + cos(a2*t)^n2)+'$')     pretty_print(html('$r=' + latex(b+sin(a1*t)^n1 + cos(a2*t)^n2)+'$'))
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{{{ {{{#!sagecell
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    except TypeError, msg:
        print msg[-200:]
        print "Unable to make sense of f,g, or a as symbolic expressions."		     except TypeError as msg:
        print(msg[-200:])
        print("Unable to make sense of f,g, or a as symbolic expressions.")
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    html('<center><font color="red">$f = %s$</font></center>'%latex(f))
    html('<center><font color="green">$g = %s$</font></center>'%latex(g))
    html('<center><font color="blue"><b>$h = %s = %s$</b></font></center>'%(lbl, latex(h)))		     pretty_print(html('<center><font color="red">$f = %s$</font></center>'%latex(f)))
    pretty_print(html('<center><font color="green">$g = %s$</font></center>'%latex(g)))
    pretty_print(html('<center><font color="blue"><b>$h = %s = %s$</b></font></center>'%(lbl, latex(h))))
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{{{ {{{#!sagecell
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                     vertical_alignment="bottom" if f(x0) < 0 else "top" )                      vertical_alignment="bottom" if f(x=x0) < 0 else "top" )
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        fi = RR(f(xi))
        fpi = RR(df(xi))		         fi = RR(f(x=xi))
        fpi = RR(df(x=xi))
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                             vertical_alignment="bottom" if f(xip1) < 0 else "top" )                              vertical_alignment="bottom" if f(x=xip1) < 0 else "top" )
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            html( t )             pretty_print(html( t ))
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{{{ {{{#!sagecell
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# polar coordinates
#(x,y)=(u*cos(v),u*sin(v)); (u_range,v_range)=([0..6],[0..2*pi,step=pi/12])

# weird example
(x,y)=(u^2-v^2,u*v+cos(u*v)); (u_range,v_range)=([-5..5],[-5..5])

thickness=4
square_length=.05
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def trans(x=input_box(u^2-v^2, label="x=",type=SR), \
         y=input_box(u*v+cos(u*v), label="y=",type=SR), \
         t_val=slider(0,10,0.2,6, label="Length of curves"), \
         u_percent=slider(0,1,0.05,label="<font color='red'>u</font>", default=.7),
         v_percent=slider(0,1,0.05,label="<font color='blue'>v</font>", default=.7),
         u_range=input_box(range(-5,5,1), label="u lines"),
         v_range=input_box(range(-5,5,1), label="v lines")):
     thickness=4
     u_val = min(u_range)+(max(u_range)-min(u_range))*u_percent
     v_val = min(v_range)+(max(v_range)-min(v_range))*v_percent
     t_min = -t_val
     t_max = t_val
     g1=sum([parametric_plot((i,v), (v,t_min,t_max), rgbcolor=(1,0,0)) for i in u_range])
     g2=sum([parametric_plot((u,i), (u,t_min,t_max), rgbcolor=(0,0,1)) for i in v_range])
     vline_straight=parametric_plot((u,v_val), (u,t_min,t_max), rgbcolor=(0,0,1), linestyle='-',thickness=thickness)
     uline_straight=parametric_plot((u_val, v), (v,t_min,t_max),rgbcolor=(1,0,0), linestyle='-',thickness=thickness)
 
     (g1+g2+vline_straight+uline_straight).save("uv_coord.png",aspect_ratio=1, figsize=[5,5], axes_labels=['$u$','$v$'])
     xuv = fast_float(x,'u','v')
     yuv = fast_float(y,'u','v')
     xvu = fast_float(x,'v','u')
     yvu = fast_float(y,'v','u')
     g3=sum([parametric_plot((partial(xuv,i),partial(yuv,i)), (v,t_min,t_max), rgbcolor=(1,0,0)) for i in u_range])
     g4=sum([parametric_plot((partial(xvu,i),partial(yvu,i)), (u,t_min,t_max), rgbcolor=(0,0,1)) for i in v_range])
     uline=parametric_plot((partial(xuv,u_val),partial(yuv,u_val)),(v,t_min,t_max),rgbcolor=(1,0,0), linestyle='-',thickness=thickness)
     vline=parametric_plot((partial(xvu,v_val),partial(yvu,v_val)), (u,t_min,t_max), rgbcolor=(0,0,1), linestyle='-',thickness=thickness)
     (g3+g4+vline+uline).save("xy_coord.png", aspect_ratio=1, figsize=[5,5], axes_labels=['$x$','$y$'])
     print jsmath("x=%s, \: y=%s"%(latex(x), latex(y)))
     print "<html><table><tr><td><img src='cell://uv_coord.png'/></td><td><img src='cell://xy_coord.png'/></td></tr></table></html>"		 def trans(x=input_box(x, label="x",type=SR),
         y=input_box(y, label="y",type=SR),
         u_percent=slider(0,1,0.05,label="u", default=.7),
         v_percent=slider(0,1,0.05,label="v", default=.7),
         t_val=slider(0,10,0.2,6, label="Length"),
         u_range=input_box(u_range, label="u lines"),
         v_range=input_box(v_range, label="v lines")):

    x(u,v)=x
    y(u,v)=y
    u_val = min(u_range)+(max(u_range)-min(u_range))*u_percent
    v_val = min(v_range)+(max(v_range)-min(v_range))*v_percent
    t_min = -t_val
    t_max = t_val
    uvplot=sum([parametric_plot((i,v), (v,t_min,t_max), color='red',axes_labels=['u','v'],figsize=[5,5]) for i in u_range])
    uvplot+=sum([parametric_plot((u,i), (u,t_min,t_max), color='blue',axes_labels=['u','v']) for i in v_range])
    uvplot+=parametric_plot((u,v_val), (u,t_min,t_max), rgbcolor=(0,0,1), linestyle='-',thickness=thickness)
    uvplot+=parametric_plot((u_val, v), (v,t_min,t_max),rgbcolor=(1,0,0), linestyle='-',thickness=thickness)
    pt=vector([u_val,v_val])
    du=vector([(t_max-t_min)*square_length,0])
    dv=vector([0,(t_max-t_min)*square_length])
    uvplot+=polygon([pt,pt+dv,pt+du+dv,pt+du],color='purple',alpha=0.7)
    uvplot+=line([pt,pt+dv,pt+du+dv,pt+du],color='green')

    T(u,v)=(x,y)
    xuv = fast_float(x,'u','v')
    yuv = fast_float(y,'u','v')
    xvu = fast_float(x,'v','u')
    yvu = fast_float(y,'v','u')
    xyplot=sum([parametric_plot((partial(xuv,i),partial(yuv,i)), (v,t_min,t_max), color='red', axes_labels=['x','y'],figsize=[5,5]) for i in u_range])
    xyplot+=sum([parametric_plot((partial(xvu,i),partial(yvu,i)), (u,t_min,t_max), color='blue') for i in v_range])
    xyplot+=parametric_plot((partial(xuv,u_val),partial(yuv,u_val)),(v,t_min,t_max),color='red', linestyle='-',thickness=thickness)
    xyplot+=parametric_plot((partial(xvu,v_val),partial(yvu,v_val)), (u,t_min,t_max), color='blue', linestyle='-',thickness=thickness)
    jacobian(u,v)=abs(T.diff().det()).simplify_full()
    t_vals=[0..1,step=t_val*.01]
    vertices=[(x(*c),y(*c)) for c in [pt+t*dv for t in t_vals]]
    vertices+=[(x(*c),y(*c)) for c in [pt+dv+t*du for t in t_vals]]
    vertices+=[(x(*c),y(*c)) for c in [pt+(1-t)*dv+du for t in t_vals]]
    vertices+=[(x(*c),y(*c)) for c in [pt+(1-t)*du for t in t_vals]]
    xyplot+=polygon(vertices,color='purple',alpha=0.7)
    xyplot+=line(vertices,color='green')
    pretty_print(html("$T(u,v)=%s$"%(latex(T(u,v)))))
    pretty_print(html("Jacobian: $%s$"%latex(jacobian(u,v))))
    pretty_print(html("A very small region in $xy$ plane is approximately %0.4g times the size of the corresponding region in the $uv$ plane"%jacobian(u_val,v_val).n()))
    show(graphics_array([uvplot,xyplot]))
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{{{ {{{#!sagecell
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dot = point((x0,f(x0)),pointsize=80,rgbcolor=(1,0,0))
@interact
def _(order=(1..12)):		 dot = point((x0,f(x=x0)),pointsize=80,rgbcolor=(1,0,0))
@interact
def _(order=[1..12]):
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    html('$f(x)\;=\;%s$'%latex(f))
    html('$\hat{f}(x;%s)\;=\;%s+\mathcal{O}(x^{%s})$'%(x0,latex(ft),order+1))		     pretty_print(html(r'$f(x)\;=\;%s$'%latex(f)))
    pretty_print(html(r'$\hat{f}(x;%s)\;=\;%s+\mathcal{O}(x^{%s})$'%(x0,latex(ft),order+1)))
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{{{
html("<h2>Limits: <i>ε-δ</i></h2>")
html("This allows you to estimate which values of <i>δ</i> guarantee that <i>f</i> is within <i>ε</i> units of a limit.")
html("<ul><li>Modify the value of <i>f</i> to choose a function.</li>")
html("<li>Modify the value of <i>a</i> to change the <i>x</i>-value where the limit is being estimated.</li>")
html("<li>Modify the value of <i>L</i> to change your guess of the limit.</li>")
html("<li>Modify the values of <i>δ</i> and <i>ε</i> to modify the rectangle.</li></ul>")
html("If the blue curve passes through the pink boxes, your values for <i>δ</i> and/or <i>ε</i> are probably wrong.")
@interact
def delta_epsilon(f = input_box(default=(x^2-x)/(x-1)), a=input_box(default=1), L = input_box(default=1), delta=input_box(label="δ",default=0.1), epsilon=input_box(label="ε",default=0.1), xm=input_box(label="<i>x</i><sub>min</sub>",default=-1), xM=input_box(label="<i>x</i><sub>max</sub>",default=4)):		 {{{#!sagecell
pretty_print(html("<h2>Limits: <i>ε-δ</i></h2>"))
pretty_print(html("This allows you to estimate which values of <i>δ</i> guarantee that <i>f</i> is within <i>ε</i> units of a limit."))
pretty_print(html("<ul><li>Modify the value of <i>f</i> to choose a function.</li>"))
pretty_print(html("<li>Modify the value of <i>a</i> to change the <i>x</i>-value where the limit is being estimated.</li>"))
pretty_print(html("<li>Modify the value of <i>L</i> to change your guess of the limit.</li>"))
pretty_print(html("<li>Modify the values of <i>δ</i> and <i>ε</i> to modify the rectangle.</li></ul>"))
pretty_print(html("If the blue curve passes through the pink boxes, your values for <i>δ</i> and/or <i>ε</i> are probably wrong."))
@interact
def delta_epsilon(f = input_box(default=(x^2-x)/(x-1), label="$f$"), a=input_box(default=1, label="$a$"), L = input_box(default=1, label="$L$"), delta=input_box(label=r"$\delta$",default=0.1), epsilon=input_box(label=r"$\varepsilon$",default=0.1), xm=input_box(label=r"$x_{min}$",default=-1), xM=input_box(label=r"$x_{max}$",default=4)):
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{{{ {{{#!sagecell
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    html('<h3>A graphical illustration of $\lim_{x -> 0} \sin(x)/x =1$</h3>')
    html('Below is the unit circle, so the length of the <font color=red>red line</font> is |sin(x)|')
    html('and the length of the <font color=blue>blue line</font> is |tan(x)| where x is the length of the arc.') 
    html('From the picture, we see that |sin(x)| $\le$ |x| $\le$ |tan(x)|.')
    html('It follows easily from this that cos(x) $\le$ sin(x)/x $\le$ 1 when x is near 0.')
    html('As $\lim_{x ->0} \cos(x) =1$, we conclude that $\lim_{x -> 0} \sin(x)/x =1$.')		     pretty_print(html(r'<h3>A graphical illustration of $\lim_{x -> 0} \sin(x)/x =1$</h3>'))
    pretty_print(html(r'Below is the unit circle, so the length of the <font color=red>red line</font> is |sin(x)|'))
    pretty_print(html(r'and the length of the <font color=blue>blue line</font> is |tan(x)| where x is the length of the arc.'))
    pretty_print(html(r'From the picture, we see that |sin(x)| $\le$ |x| $\le$ |tan(x)|.'))
    pretty_print(html(r'It follows easily from this that cos(x) $\le$ sin(x)/x $\le$ 1 when x is near 0.'))
    pretty_print(html(r'As $\lim_{x ->0} \cos(x) =1$, we conclude that $\lim_{x -> 0} \sin(x)/x =1$.'))
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{{{ {{{#!sagecell
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def quads(q = selector(quadrics.keys()), a = slider(0,5,1/2,default = 1)): def quads(q = selector(list(quadrics)), a = slider(0,5,1/2,default = 1)):
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    if a==0 or q=='Cone': html('<center>$'+latex(f)+' \ $'+ '(degenerate)</center>')
    else: html('<center>$'+latex(f)+'$ </center>')		     if a==0 or q=='Cone': pretty_print(latex(f), "   (degenerate)")
    else: pretty_print(latex(f))
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{{{ {{{#!sagecell
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sin,cos = math.sin,math.cos
html("<h1>The midpoint rule for a function of two variables</h1>")
pretty_print(html(r"<h1>The midpoint rule for a function of two variables</h1>"))
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    html("$$\int_{"+str(R16(y_start))+"}^{"+str(R16(y_end))+"} "+ "\int_{"+str(R16(x_start))+"}^{"+str(R16(x_end))+"} "+func+"\ dx \ dy$$")
    html('<p style="text-align: center;">Numerical approximation: ' + str(num_approx)+'</p>')		     pretty_print(html(r"$\int_{"+str(R16(y_start))+r"}^{"+str(R16(y_end))+r"} "+ r"\int_{"+str(R16(x_start))+r"}^{"+str(R16(x_end))+r"} "+latex(SR(func))+r"\ dx \ dy$"))
    pretty_print(html(r'<p style="text-align: center;">Numerical approximation: ' + str(num_approx)+r'</p>'))
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{{{
from scipy.special.orthogonal import p_roots		 {{{#!sagecell
import scipy
import numpy
from scipy.special.orthogonal import p_roots, t_roots, u_roots
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from numpy import linspace from numpy import linspace, asanyarray, diff
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            'Chebyshev': {'w': 1/sqrt(1-x**2), 'xmin': -1, 'xmax': 1, 'func': t_roots},
                'Chebyshev2': {'w': sqrt(1-x**2), 'xmin': -1, 'xmax': 1, 'func': u_roots},
                'Trapezoid': {'w': 1, 'xmin': -1, 'xmax': 1, 'func': lambda n: (linspace(-1r,1,n), numpy.array([1.0r]+[2.0r]*(n-2)+[1.0r])*1.0r/n)},
                'Simpson': {'w': 1, 'xmin': -1, 'xmax': 1, 'func': lambda n: (linspace(-1r,1,n), numpy.array([1.0r]+[4.0r,2.0r]*int((n-3.0r)/2.0r)+[4.0r,1.0r])*2.0r/(3.0r*n))}}		      'Chebyshev': {'w': 1/sqrt(1-x**2), 'xmin': -1, 'xmax': 1, 'func': t_roots},
     'Chebyshev2': {'w': sqrt(1-x**2), 'xmin': -1, 'xmax': 1, 'func': u_roots},
     'Trapezoid': {'w': 1, 'xmin': -1, 'xmax': 1,          'func': lambda n: (linspace(-1r,1,n), numpy.array([1.0r]+[2.0r]*(n-2)+[1.0r])*1.0r/n)},
     'Simpson': {'w': 1, 'xmin': -1, 'xmax': 1,          'func': lambda n: (linspace(-1r,1,n),
            numpy.array([1.0r]+[4.0r,2.0r]*int((n-3.0r)/2.0r)+[4.0r,1.0r])*2.0r/(3.0r*n))}}
Line 650: Line 676:
    return polygon([(center-width2,0),(center+width2,0),(center+width2,height),(center-width2,height)],**kwds)     return polygon([(center-width2,0),
        (center+width2,0),(center+width2,height),(center-width2,height)],**kwds)
Line 654: Line 681:
def weights(n=slider(1,30,1,default=10),f=input_box(default=3*x+cos(10*x)),show_method=["Legendre", "Chebyshev", "Chebyshev2", "Trapezoid","Simpson"]): def weights(n=slider(1,30,1,default=10),f=input_box(default=3*x+cos(10*x),type=SR),
    show_method=["Legendre", "Chebyshev", "Chebyshev2", "Trapezoid","Simpson"]):
Line 663: Line 691:
    scaled_ff = fast_float(scaled_func)     scaled_ff = fast_float(scaled_func, 'x')
Line 671: Line 699:
    stems = sum(line([(x,0),(x,scaled_ff(x))],rgbcolor=(1-y,1-y,1-y),thickness=2,markersize=6,alpha=y) for x,y in coords_scaled)
    points = sum([point([(x,0),(x,scaled_ff(x))],rgbcolor='black',pointsize=30) for x,_ in coords])		     stems = sum(line([(x,0),(x,scaled_ff(x))],rgbcolor=(1-y,1-y,1-y),
        thickness=2,markersize=6,alpha=y) for x,y in coords_scaled)
    points = sum([point([(x,0),
        (x,scaled_ff(x))],rgbcolor='black',pointsize=30) for x,_ in coords])
Line 677: Line 707:
    show(graph,xmin=plot_min,xmax=plot_max)     show(graph,xmin=plot_min,xmax=plot_max,aspect_ratio="auto")
Line 682: Line 712:
    y_val = map(scaled_ff,x_val)     y_val = [*map(scaled_ff,x_val)]
Line 685: Line 715:
    html("$$\sum_{i=1}^{i=%s}w_i\left(%s\\right)= %s\\approx %s =\int_{-1}^{1}%s \,dx$$"%(n,latex(f.subs(x="x_i")), approximation, integral, latex(scaled_func)))     pretty_print(html(r"$$\sum_{i=1}^{i=%s}w_i\left(%s\right)= %s\approx %s =\int_{-1}^{1}%s \,dx$$"%(n,
        latex(f), approximation, integral, latex(scaled_func))))
Line 687: Line 718:
    print "Trapezoid: %s, Simpson: %s, \nMethod: %s, Real: %s"%tuple(error_data)     print("Trapezoid: %s, Simpson: %s, \nMethod: %s, Real: %s" % tuple(error_data))
Line 698: Line 729:
{{{ {{{#!sagecell
Line 725: Line 756:
path = parametric_plot( position(t).list(), (t, start, stop), color = "black" ) path = parametric_plot( position.list(), (t, start, stop), color = "black" )
Line 729: Line 760:
velocity = derivative( position(t) )
acceleration = derivative(velocity(t))		 velocity = derivative(position, t)
acceleration = derivative(velocity, t)
Line 732: Line 763:
speed_deriv = derivative(speed) speed_deriv = derivative(speed, t)
Line 734: Line 765:
dT = derivative(tangent(t)) dT = derivative(tangent, t)
Line 755: Line 786:
    pos_tzero = position(t0)     pos_tzero = position(t=t0)
Line 759: Line 790:
    speed_component = speed(t0)
    tangent_component = speed_deriv(t0)
    normal_component = sqrt( acceleration(t0).norm()^2 - tangent_component^2 )		     speed_component = speed(t=t0)
    tangent_component = speed_deriv(t=t0)
    normal_component = sqrt( acceleration(t=t0).norm()^2 - tangent_component^2 )
Line 767: Line 798:
    tan = arrow(pos_tzero, pos_tzero + tangent(t0), rgbcolor=(0,1,0) )
    vel = arrow(pos_tzero, pos_tzero + velocity(t0), rgbcolor=(0,0.5,0))
    nor = arrow(pos_tzero, pos_tzero + normal(t0), rgbcolor=(0.5,0,0))
    acc = arrow(pos_tzero, pos_tzero + acceleration(t0), rgbcolor=(1,0,1))
    tancomp = arrow(pos_tzero, pos_tzero + tangent_component*tangent(t0), rgbcolor=(1,0,1) )
    norcomp = arrow(pos_tzero, pos_tzero + normal_component*normal(t0), rgbcolor=(1,0,1))		     tan = arrow(pos_tzero, pos_tzero + tangent(t=t0), rgbcolor=(0,1,0) )
    vel = arrow(pos_tzero, pos_tzero + velocity(t=t0), rgbcolor=(0,0.5,0))
    nor = arrow(pos_tzero, pos_tzero + normal(t=t0), rgbcolor=(0.5,0,0))
    acc = arrow(pos_tzero, pos_tzero + acceleration(t=t0), rgbcolor=(1,0,1))
    tancomp = arrow(pos_tzero, pos_tzero + tangent_component*tangent(t=t0), rgbcolor=(1,0,1) )
    norcomp = arrow(pos_tzero, pos_tzero + normal_component*normal(t=t0), rgbcolor=(1,0,1))
Line 794: Line 825:
    print "Position vector defined as r(t)=", position(t)
    print "Speed is ", N(speed(t0))
    print "Curvature is ", N(curvature)		     print("Position vector defined as r(t)={}".format(position))
    print("Speed is {}".format(N(speed(t=t0))))
    print("Curvature is {}".format(N(curvature)))
Line 810: Line 841:
{{{ {{{#!sagecell
Line 826: Line 857:
assume(t, 'real')
Line 843: Line 875:
path = parametric_plot3d( position(t).list(), (t, start, stop), color = "black" ) path = parametric_plot3d( position.list(), (t, start, stop), color = "black" )
Line 847: Line 879:
velocity = derivative( position(t) )
acceleration = derivative(velocity(t))		 velocity = derivative( position, t)
acceleration = derivative(velocity, t)
Line 850: Line 882:
speed_deriv = derivative(speed) speed_deriv = derivative(speed, t)
Line 852: Line 884:
dT = derivative(tangent(t)) dT = derivative(tangent, t)
Line 855: Line 887:
## dB = derivative(binormal(t)) ## dB = derivative(binormal, t)
Line 876: Line 908:
    pos_tzero = position(t0)     pos_tzero = position(t=t0)
Line 880: Line 912:
    speed_component = speed(t0)
    tangent_component = speed_deriv(t0)
    normal_component = sqrt( acceleration(t0).norm()^2 - tangent_component^2 )		     speed_component = speed(t=t0)
    tangent_component = speed_deriv(t=t0)
    normal_component = sqrt( acceleration(t=t0).norm()^2 - tangent_component^2 )
Line 889: Line 921:
    tan = arrow3d(pos_tzero, pos_tzero + tangent(t0), rgbcolor=(0,1,0) )
    vel = arrow3d(pos_tzero, pos_tzero + velocity(t0), rgbcolor=(0,0.5,0))
    nor = arrow3d(pos_tzero, pos_tzero + normal(t0), rgbcolor=(0.5,0,0))
    bin = arrow3d(pos_tzero, pos_tzero + binormal(t0), rgbcolor=(0,0,0.5))
    acc = arrow3d(pos_tzero, pos_tzero + acceleration(t0), rgbcolor=(1,0,1))
    tancomp = arrow3d(pos_tzero, pos_tzero + tangent_component*tangent(t0), rgbcolor=(1,0,1) )
    norcomp = arrow3d(pos_tzero, pos_tzero + normal_component*normal(t0), rgbcolor=(1,0,1))		     tan = arrow3d(pos_tzero, pos_tzero + tangent(t=t0), rgbcolor=(0,1,0) )
    vel = arrow3d(pos_tzero, pos_tzero + velocity(t=t0), rgbcolor=(0,0.5,0))
    nor = arrow3d(pos_tzero, pos_tzero + normal(t=t0), rgbcolor=(0.5,0,0))
    bin = arrow3d(pos_tzero, pos_tzero + binormal(t=t0), rgbcolor=(0,0,0.5))
    acc = arrow3d(pos_tzero, pos_tzero + acceleration(t=t0), rgbcolor=(1,0,1))
    tancomp = arrow3d(pos_tzero, pos_tzero + tangent_component*tangent(t=t0), rgbcolor=(1,0,1) )
    norcomp = arrow3d(pos_tzero, pos_tzero + normal_component*normal(t=t0), rgbcolor=(1,0,1))
Line 919: Line 951:
    print "Position vector: r(t)=", position(t)
    print "Speed is ", N(speed(t0))
    print "Curvature is ", N(curvature)
    ## print "Torsion is ", N(torsion)
    print
    print "Right-click on graphic to zoom to 400%"
    print "Drag graphic to rotate"		     print("Position vector: r(t)=", position)
    print("Speed is ", N(speed(t=t0)))
    print("Curvature is ", N(curvature))
    ## print("Torsion is ", N(torsion))
    print()
    print("Right-click on graphic to zoom to 400%")
    print("Drag graphic to rotate")
Line 936: Line 968:
{{{ http://sagenb.mc.edu/home/pub/97/

{{{#!sagecell
Line 948: Line 982:
var('x,y,z')
Rmin=1/10
Line 950: Line 985:
@interact
def _(f=input_box(default=(x^3-y^3)/(x^2+y^2)),R=slider(0.1/10,Rmax,1/10,2),x0=(0),y0=(0)):		 @interact(layout=dict(top=[['f'],['x0'],['y0']],
bottom=[['in_3d','curves','R','graphjmol']]))
def _(f=input_box((x^2-y^2)/(x^2+y^2),width=30,label='$f(x)$'),
        R=slider(Rmin,Rmax,1/10,Rmax,label=',   $R$'),
        x0=input_box(0,width=10,label='$x_0$'),
        y0=input_box(0,width=10,label='$y_0$'),
        curves=checkbox(default=false,label='Show curves'),
        in_3d=checkbox(default=false,label='3D'),
        graphjmol=checkbox(default=true,label='Interactive graph')):
    if graphjmol:
        view_method = 'jmol'
    else:
        view_method = 'tachyon'
Line 959: Line 1005:
    
Line 961: Line 1007:
    limit = plot3d(g,(t,0,2*pi),(r,1/100,R),transformation=cylinder,rgbcolor=(0,1,0))     collapsing_surface = plot3d(g,(t,0,2*pi),(r,1/100,R),transformation=cylinder,rgbcolor=(0,1,0))
Line 963: Line 1009:
    show(surface+limit)
    print html('Enter $(x_0 ,y_0 )$ above and see what happens as R approaches zero.')
    print html('The surface has a limit as $(x,y)$ approaches ('+str(x0)+','+str(y0)+') if the green region collapses to a point')		     G = surface+collapsing_surface
    pretty_print(html('Enter $(x_0 ,y_0 )$ above and see what happens as $ R \\rightarrow 0 $.'))
    pretty_print(html('The surface has a limit as $(x,y) \\rightarrow $ ('+str(x0)+','+str(y0)+') if the green region collapses to a point.'))

# If checked, add a couple of curves on the surface corresponding to limit as x->x0 for y=x^(3/5),
# and as y->y0 for x=y^(3/5). Should make this more robust but perhaps using
# these relatively obtuse curves could eliminate problems.

    if curves:
        curve_x = parametric_plot3d([x0-t,y0-t^(3/5),f(x=x0-t,y=y0-t^(3/5))],(t,Rmin,Rmax),color='red',thickness=10)
        curve_y = parametric_plot3d([x0+t^(3/5),y0+t,f(x=x0+t^(3/5),y=y0+t)],(t,Rmin,Rmax),color='red',thickness=10)
        R2 = Rmin/4
        G += arrow((x0-Rmin,y0-Rmin^(3/5),f(x=x0-Rmin,y=y0-Rmin^(3/5))),(x0-R2,y0-R2^(3/5),f(x=x0-R2,y=y0-R2^(3/5))),size=30 )
        G += arrow((x0+Rmin^(3/5),y0+Rmin,f(x=x0+Rmin^(3/5),y=y0+Rmin)),(x0+R2^(3/5),y0+R2,f(x=x0+R2^(3/5),y=y0+R2)),size=30 )

        limit_x = limit(f(x=x0-t,y=y0-t^(3/5)),t=0)
        limit_y = limit(f(x=x0+t^(3/5),y=y0+t),t=0)
        text_x = text3d(limit_x,(x0,y0,limit_x))
        text_y = text3d(limit_y,(x0,y0,limit_y))
        G += curve_x+curve_y+text_x+text_y
 
    
        pretty_print(html('The red curves represent a couple of trajectories on the surface. If they do not meet, then'))
        pretty_print(html('there is also no limit. (If computer hangs up, likely the computer can not do these limits.)'))
        pretty_print(html(r'<center><font color="red">$\lim_{(x,?)\rightarrow(x_0,y_0)} f(x,y) =%s$</font>'%str(limit_x)+r' and <font color="red">$\lim_{(?,y)\rightarrow(x_0,y_0)} f(x,y) =%s$</font></center>'%str(limit_y)))
        
    if in_3d:
        show(G,stereo="redcyan",viewer=view_method)
    else:
        show(G,perspective_depth=true,viewer=view_method)
Line 970: Line 1043:
{{{ {{{#!sagecell
Line 987: Line 1060:
Rmax=2
@interact
def _(f=input_box(default=(x^3-y^3)/(x^2+y^2)),
      N=slider(5,100,1,10,label='Number of Contours'),
      x0=(0),y0=(0)):

    print html('Enter $(x_0 ,y_0 )$ above and see what happens as the number of contour levels increases.')
    print html('A surface will have a limit in the center of this graph provided there is not a sudden change in color there.') 		var('x,y,z,u')
@interact(layout=dict(top=[['f'],['x0'],['y0']],
bottom=[['N'],['R']]))
def _(f=input_box(default=(x*y^2)/(x^2+y^4),width=30,label='$f(x)$'),
        N=slider(5,100,1,10,label='Number of Contours'),
        R=slider(0.1,1,0.01,1,label='Radius of circular neighborhood'),
        x0=input_box(0,width=10,label='$x_0$'),
        y0=input_box(0,width=10,label='$y_0$')):

    pretty_print(html(r'Enter $(x_0 ,y_0 )$ above and see what happens as the number of contour levels $\rightarrow \infty $.'))
    pretty_print(html('A surface will have a limit in the center of this graph provided there is not a sudden change in color there.'))

# Need to make certain the min and max contour lines are not huge due to asymptotes. If so, clip and start contours at some reasonable
# values so that there are a nice collection of contours to show around the desired point.
Line 997: Line 1076:
    surface += parametric_plot([R*cos(u),R*sin(u)],[0,2*pi],color='black')
# Nice to use if f=x*y^2/(x^2 + y^4)
# var('u')
# surface += parametric_plot([u^2,u],[u,-1,1],color='black')
Line 998: Line 1081:
    show(limit_point+surface)}}} # show(limit_point+surface)
    show(surface)
}}}
Line 1007: Line 1092:
{{{ {{{#!sagecell
Line 1060: Line 1145:
{{{
%hide
%auto		 {{{#!sagecell
Line 1077: Line 1160:
 html(r'Function $ f(x,y)=%s$ '%latex(f(x,y)))  pretty_print(html(r'Function $ f(x,y)=%s$ '%latex(f(x,y))))
Line 1091: Line 1174:
              html(r'<tr><td>$\quad f(%s,%s)\quad $</td><td>$\quad %s$</td>\
              </tr>'%(latex(x0),latex(y0),z0.n()))		               pretty_print(html(r'<tr><td>$\quad f(%s,%s)\quad $</td><td>$\quad %s$</td>\
              </tr>'%(latex(x0),latex(y0),z0.n())))
Line 1121: Line 1204:
{{{ {{{#!sagecell
Line 1125: Line 1208:
html('Points x0 and y0 are values where the exact value of the function \ pretty_print(html('Points x0 and y0 are values where the exact value of the function \
Line 1127: Line 1210:
and approximation by differential at shifted point are compared.') and approximation by differential at shifted point are compared.'))
Line 1145: Line 1228:
  html(r'Function $ f(x,y)=%s \approx %s $ '%(latex(f(x,y)),latex(tangent(x,y))))
  html(r' $f %s = %s$'%(latex((x0,y0)),latex(exact_value_ori)))
  html(r'Shifted point $%s$'%latex(((x0+deltax),(y0+deltay))))
  html(r'Value of the function in shifted point is $%s$'%f(x0+deltax,y0+deltay))
  html(r'Value on the tangent plane in shifted point is $%s$'%latex(approx_value))
  html(r'Error is $%s$'%latex(abs_error)) 		   pretty_print(html(r'Function $ f(x,y)=%s \approx %s $ '%(latex(f(x,y)),latex(tangent(x,y)))))
  pretty_print(html(r' $f %s = %s$'%(latex((x0,y0)),latex(exact_value_ori))))
  pretty_print(html(r'Shifted point $%s$'%latex(((x0+deltax),(y0+deltay)))))
  pretty_print(html(r'Value of the function in shifted point is $%s$'%f(x0+deltax,y0+deltay)))
  pretty_print(html(r'Value on the tangent plane in shifted point is $%s$'%latex(approx_value)))
  pretty_print(html(r'Error is $%s$'%latex(abs_error)))
Line 1159: Line 1242:
{{{ {{{#!sagecell
Line 1168: Line 1251:
      order=(1..10)):       order=[1..10]):
Line 1187: Line 1270:
    html('$F(x,y) = e^{-(x^2+y^2)/2} \\cos(y) \\sin(x^2+y^2)$')     pretty_print(html('$F(x,y) = e^{-(x^2+y^2)/2} \\cos(y) \\sin(x^2+y^2)$'))
Line 1197: Line 1280:
{{{ https://cloud.sagemath.com/projects/19575ea0-317e-402b-be57-368d04c113db/files/pub/2801-2901/2829.sagews

{{{#!sagecell
Line 1201: Line 1286:
##
##
## An updated version of this worksheet may be available at http://sagenb.mc.edu
Line 1299: Line 1387:
{{{ http://sagenb.mc.edu/home/pub/89/

Note that this works in Sage cell, but causes a zip file error in Jupyter

{{{#!sagecell
Line 1304: Line 1396:
##
Line 1306: Line 1399:
@interact
def _(f=input_box(default=6-4*x^2-y^2*2/5,label='$f(x,y) = $'),
        g=input_box(default=-2+sin(x)+sin(y),label='$g(x,y) = $'),
        u=input_box(default=cos(t),label='$u(t) = $'),
        v=input_box(default=2*sin(t),label='$v(t) = $'),
        a=input_box(default=0,label='$a = $'),
        b=input_box(default=3*pi/2,label='$b = $'),		 @interact(layout=dict(top=[['f','u'],['g','v']],
left=[['a'],['b'],['in_3d'],['smoother']],
bottom=[['xx','yy']]))
def _(f=input_box(default=6-4*x^2-y^2*2/5,label='Top = $f(x,y) = $',width=30),
        g=input_box(default=-2+sin(x)+sin(y),label='Bottom = $g(x,y) = $',width=30),
        u=input_box(default=cos(t),label='   $ x = u(t) = $',width=20),
        v=input_box(default=2*sin(t),label='   $ y = v(t) = $',width=20),
        a=input_box(default=0,label='$a = $',width=10),
        b=input_box(default=3*pi/2,label='$b = $',width=10),
Line 1315: Line 1410:
        smoother=checkbox(default=false)):         in_3d = checkbox(default=true,label='3D'),
 smoother=checkbox(default=false),
        auto_update=true):
Line 1317: Line 1414:
    ds = sqrt(derivative(u(t),t)^2+derivative(v(t),t)^2)     ds = sqrt(derivative(u,t)^2+derivative(v,t)^2)
Line 1321: Line 1418:
    A = (f(x=u(t),y=v(t))-g(x=u(t),y=v(t)))*ds.simplify_trig().simplify()     A = (f(x=u,y=v)-g(x=u,y=v))*ds.simplify_trig().simplify()
Line 1326: Line 1423:
    line_integral = integral(A,t,a,b) # If you want Sage to try, uncomment the lines below.

# line_integral = integrate(A,t,a,b)
# html(r'<align=center size=+1>Lateral Surface Area = $ %s $ </font>'%latex(line_integral))
Line 1328: Line 1429:
       
    html(r'<h4 align=center>Lateral Surface Area = $ %s $ </h4>'%latex(line_integral))

    html(r'<h4 align=center>Lateral Surface $ \approx $ %s</h2>'%str(line_integral_approx))
    pretty_print(html(r'<font align=center size=+1>Lateral Surface $ \approx $ %s</font>'%str(line_integral_approx)))
Line 1338: Line 1437:
    G += parametric_plot3d([u,v,g(x=u(t),y=v(t))],(t,a,b),thickness=2,color='red')
    G += parametric_plot3d([u,v,f(x=u(t),y=v(t))],(t,a,b),thickness=2,color='red')		     G += parametric_plot3d([u,v,g(x=u,y=v)],(t,a,b),thickness=2,color='red')
    G += parametric_plot3d([u,v,f(x=u,y=v)],(t,a,b),thickness=2,color='red')
Line 1348: Line 1447:
        G += parametric_plot3d([u(w),v(w),s*f(x=u(w),y=v(w))+(1-s)*g(x=u(w),y=v(w))],(s,0,1),thickness=lat_thick,color='yellow',opacity=0.9)
    show(G,spin=true)		         G += parametric_plot3d([u(t=w),v(t=w),s*f(x=u(t=w),y=v(t=w))+(1-s)*g(x=u(t=w),y=v(t=w))],(s,0,1),thickness=lat_thick,color='yellow',opacity=0.9)
             if in_3d:
        show(G,stereo='redcyan',spin=true)
    else:
        show(G,perspective_depth=true,spin=true)
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{{{
Note that this works in Sage cell, but causes a zip file error in Jupyter.
{{{#!sagecell
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{{{ https://cloud.sagemath.com/projects/19575ea0-317e-402b-be57-368d04c113db/files/pub/2801-2901/2827-$%20%5Cint_%7BC%7D%20%5Cleft%20%5Clangle%20M,N,P%20%5Cright%20%5Crangle%20dr%20$%20=%20$%20%25s%20$.sagews

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## An updated version of this worksheet may be available at http://sagenb.mc.edu
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    u(t) = u
    v(t) = v
    w(t) = w
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    html(r'<h2 align=center>$ \int_{C} \left \langle M,N,P \right \rangle dr $ = $ %s $ </h2>'%latex(line_integral))     pretty_print(html(r'<h2 align=center>$ \int_{C} \left \langle M,N,P \right \rangle dr $ = $ %s $ </h2>'%latex(line_integral)))

Sage Interactions - Calculus

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Contents

  1. Sage Interactions - Calculus
    1. Root Finding Using Bisection
    2. Newton's Method
    3. A contour map and 3d plot of two inverse distance functions
    4. A simple tangent line grapher
    5. Numerical integrals with the midpoint rule
    6. Numerical integrals with various rules
    7. Some polar parametric curves
    8. Function tool
    9. Newton-Raphson Root Finding
    10. Coordinate Transformations
    11. Taylor Series
    12. Illustration of the precise definition of a limit
    13. A graphical illustration of sin(x)/x -> 1 as x-> 0
    14. Quadric Surface Plotter
    15. The midpoint rule for numerically integrating a function of two variables
    16. Gaussian (Legendre) quadrature
    17. Vector Calculus, 2-D Motion
    18. Vector Calculus, 3-D Motion
    19. Multivariate Limits by Definition
    20. Directional Derivatives
    21. 3D graph with points and curves
    22. Approximating function in two variables by differential
    23. Taylor approximations in two variables
    24. Volumes over non-rectangular domains
    25. Lateral Surface Area
    26. Parametric surface example
    27. Line Integrals in 3D Vector Field

Root Finding Using Bisection

by William Stein

bisect.png

Newton's Method

Note that there is a more complicated Newton's method below.

by William Stein

https://cloud.sagemath.com/projects/19575ea0-317e-402b-be57-368d04c113db/files/pub/2801-2901/2824-Double%20Precision%20Root%20Finding%20Using%20Newton's%20Method.sagews

newton.png

A contour map and 3d plot of two inverse distance functions

by William Stein

https://cloud.sagemath.com/projects/19575ea0-317e-402b-be57-368d04c113db/files/pub/2801-2901/2823.sagews

mountains.png

A simple tangent line grapher

by Marshall Hampton

tangents.png

Numerical integrals with the midpoint rule

by Marshall Hampton

num_int.png

Numerical integrals with various rules

by Nick Alexander (based on the work of Marshall Hampton)

num_int2.png

Some polar parametric curves

by Marshall Hampton. This is not very general, but could be modified to show other families of polar curves.

polarcurves1.png

Function tool

Enter symbolic functions f, g, and a, a range, then click the appropriate button to compute and plot some combination of f, g, and a along with f and g. This is inspired by the Matlab funtool GUI.

funtool.png

Newton-Raphson Root Finding

by Neal Holtz

This allows user to display the Newton-Raphson procedure one step at a time. It uses the heuristic that, if any of the values of the controls change, then the procedure should be re-started, else it should be continued.

newtraph.png

Coordinate Transformations

by Jason Grout

coordinate-transform-1.png coordinate-transform-2.png

Taylor Series

by Harald Schilly

taylor_series_animated.gif

Illustration of the precise definition of a limit

by John Perry

I'll break tradition and put the image first. Apologies if this is Not A Good Thing.

snapshot_epsilon_delta.png

A graphical illustration of sin(x)/x -> 1 as x-> 0

by Wai Yan Pong

sinelimit.png

Quadric Surface Plotter

by Marshall Hampton. This is pretty simple, so I encourage people to spruce it up. In particular, it isn't set up to show all possible types of quadrics.

quadrics.png

The midpoint rule for numerically integrating a function of two variables

by Marshall Hampton

numint2d.png

Gaussian (Legendre) quadrature

by Jason Grout

The output shows the points evaluated using Gaussian quadrature (using a weight of 1, so using Legendre polynomials). The vertical bars are shaded to represent the relative weights of the points (darker = more weight). The error in the trapezoid, Simpson, and quadrature methods is both printed out and compared through a bar graph. The "Real" error is the error returned from scipy on the definite integral.

quadrature1.png quadrature2.png

Vector Calculus, 2-D Motion

By Rob Beezer

A fast_float() version is available in a worksheet

motion2d.png

Vector Calculus, 3-D Motion

by Rob Beezer

Available as a worksheet

motion3d.png

Multivariate Limits by Definition

by John Travis

http://sagenb.mc.edu/home/pub/97/

3D_Limit_Defn.png

3D_Limit_Defn_Contours.png

Directional Derivatives

This interact displays graphically a tangent line to a function, illustrating a directional derivative (the slope of the tangent line).

directional derivative.png

3D graph with points and curves

By Robert Marik

This sagelet is handy when showing local, constrained and absolute maxima and minima in two variables. Available as a worksheet

3Dgraph_with_points.png

Approximating function in two variables by differential

by Robert Marik

3D_differential.png

Taylor approximations in two variables

by John Palmieri

This displays the nth order Taylor approximation, for n from 1 to 10, of the function sin(x2 + y2) cos(y) exp(-(x2+y2)/2).

taylor-3d.png

Volumes over non-rectangular domains

by John Travis

https://cloud.sagemath.com/projects/19575ea0-317e-402b-be57-368d04c113db/files/pub/2801-2901/2829.sagews

3D_Irregular_Volume.png

Lateral Surface Area

by John Travis

http://sagenb.mc.edu/home/pub/89/

Note that this works in Sage cell, but causes a zip file error in Jupyter

Lateral_Surface.png

Parametric surface example

by Marshall Hampton

Note that this works in Sage cell, but causes a zip file error in Jupyter.

parametric_surface.png

Line Integrals in 3D Vector Field

by John Travis

https://cloud.sagemath.com/projects/19575ea0-317e-402b-be57-368d04c113db/files/pub/2801-2901/2827-$%20%5Cint_%7BC%7D%20%5Cleft%20%5Clangle%20M,N,P%20%5Cright%20%5Crangle%20dr%20$%20=%20$%20%25s%20$.sagews

3D_Line_Integral.png

interact/calculus (last edited 2020-08-11 14:10:09 by kcrisman)