Differences between revisions 41 and 111 (spanning 70 versions)
⇤ ← Revision 41 as of 2011-06-17 16:45:44 →
Size: 57292
Editor: JohnTravis
Comment:
← Revision 111 as of 2020-06-02 14:13:05 → ⇥
Size: 62966
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])		     min_y = min(0, find_local_minimum(func,a,b)[0])
    max_y = max(0, find_local_maximum(func,a,b)[0])
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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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            '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 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("$$\sum_{i=1}^{i=%s}w_i\left(%s\\right)= %s\\approx %s =\int_{-1}^{1}%s \,dx$$"%(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://www.sagenb.org/home/pub/2828/

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

{{{#!sagecell
Line 946: Line 978:
## An updated version of this worksheet may be available at http://sagenb.mc.edu
Line 951: Line 982:
var('x,y,z')
Rmin=1/10
Line 953: 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 962: Line 1005:
    
Line 964: 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 966: 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('\n<center><font color="red">$\lim_{(x,?)\\rightarrow(x_0,y_0)} f(x,y) =%s$</font>'%str(limit_x)+' 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 973: Line 1043:
{{{ {{{#!sagecell
Line 990: Line 1061:
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('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 1000: Line 1077:
    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 1001: Line 1082:
    show(limit_point+surface)}}} # show(limit_point+surface)
    pretty_print(table([[surface],['hi']]))
}}}
Line 1010: Line 1093:
{{{ {{{#!sagecell
Line 1063: Line 1146:
{{{
%hide
%auto		 {{{#!sagecell
Line 1080: Line 1161:
 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 1094: Line 1175:
              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 1124: Line 1205:
{{{ {{{#!sagecell
Line 1128: Line 1209:
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 1130: Line 1211:
and approximation by differential at shifted point are compared.') and approximation by differential at shifted point are compared.'))
Line 1148: Line 1229:
  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 1162: Line 1243:
{{{ {{{#!sagecell
Line 1171: Line 1252:
      order=(1..10)):       order=[1..10]):
Line 1190: Line 1271:
    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 1200: Line 1281:
http://www.sagenb.org/home/pub/2829/

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

{{{#!sagecell
Line 1303: Line 1384:
== Lateral Surface Area == == Lateral Surface Area (FIXME in Jupyter) ==
Line 1307: Line 1388:
http://www.sagenb.org/home/pub/2826/

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

{{{#!sagecell
Line 1314: Line 1395:
##
Line 1316: Line 1398:
@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 1325: Line 1409:
        smoother=checkbox(default=false)):         in_3d = checkbox(default=true,label='3D'),
 smoother=checkbox(default=false),
        auto_update=true):
Line 1327: Line 1413:
    ds = sqrt(derivative(u(t),t)^2+derivative(v(t),t)^2)     ds = sqrt(derivative(u,t)^2+derivative(v,t)^2)
Line 1331: Line 1417:
    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 1336: Line 1422:
    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 1338: Line 1428:
       
    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 1348: Line 1436:
    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 1358: Line 1446:
        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)
Line 1364: Line 1456:
== Parametric surface example == == Parametric surface example (FIXME in Jupyter) ==
Line 1366: Line 1458:
{{{ {{{#!sagecell
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http://www.sagenb.org/home/pub/2827/

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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 (FIXME in Jupyter)
    26. Parametric surface example (FIXME in Jupyter)
    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 (FIXME in Jupyter)

by John Travis

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

Lateral_Surface.png

Parametric surface example (FIXME in Jupyter)

by Marshall Hampton

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)