Differences between revisions 39 and 58 (spanning 19 versions)

Sage Interactions - Calculus

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Sage Interactions - Calculus

Root Finding Using Bisection

by William Stein

Newton's Method

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

by William Stein

http://sagenb.org/home/pub/2824/

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

by William Stein

http://sagenb.org/home/pub/2823/

A simple tangent line grapher

by Marshall Hampton

Numerical integrals with the midpoint rule

by Marshall Hampton

Numerical integrals with various rules

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

Some polar parametric curves

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

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.

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.

Coordinate Transformations

by Jason Grout

Taylor Series

by Harald Schilly

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.

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

by Wai Yan Pong

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.

The midpoint rule for numerically integrating a function of two variables

by Marshall Hampton

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.

Vector Calculus, 2-D Motion FIXME

By Rob Beezer

A fast_float() version is available in a worksheet

Vector Calculus, 3-D Motion

by Rob Beezer

Available as a worksheet

Multivariate Limits by Definition FIXME

by John Travis

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

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

Approximating function in two variables by differential

by Robert Marik

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).

Volumes over non-rectangular domains

by John Travis

http://www.sagenb.org/home/pub/2829/

Lateral Surface Area

by John Travis

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

Parametric surface example

by Marshall Hampton

Line Integrals in 3D Vector Field

by John Travis

http://www.sagenb.org/home/pub/2827/

-  ⇤ ← Revision 39 as of 2011-06-16 18:15:15 → 
  Size: 45797
  Editor: was
  Comment:
+   ← Revision 58 as of 2013-04-24 18:48:01 → ⇥
  Size: 61047
  Editor: travis
  Comment:
-Deletions are marked like this.
+Additions are marked like this.
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+Line 10:
-{{{
+{{{#!sagecell
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+{{{#!sagecell
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+{{{#!sagecell
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+{{{#!sagecell
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+{{{#!sagecell
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+{{{#!sagecell
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+{{{#!sagecell
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+Line 457:
+# 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
-Line 459:
+Line 469:
-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"), \
+def trans(x=input_box(x, label="x",type=SR),
         y=input_box(y, label="y",type=SR),
-Line 464:
+Line 473:
-         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>"
}}}
+         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=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')
    html("$T(u,v)=%s$"%(latex(T(u,v))))
    html("Jacobian: $%s$"%latex(jacobian(u,v)))
    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())
    html.table([[uvplot,xyplot]])}}}
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+Line 521:
-{{{
+{{{#!sagecell
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+Line 544:
-{{{
+{{{#!sagecell
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+Line 571:
-{{{
+{{{#!sagecell
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-{{{
+{{{#!sagecell
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+Line 612:
-{{{
+{{{#!sagecell
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+Line 658:
-{{{
from scipy.special.orthogonal import p_roots
+{{{#!sagecell
import scipy
import numpy
from scipy.special.orthogonal import p_roots, t_roots, u_roots
-Line 643:
+Line 670:
-                '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 680:
-    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 685:
-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 695:
-    scaled_ff = fast_float(scaled_func)
+    scaled_ff = fast_float(scaled_func, 'x')
-Line 671:
+Line 703:
-    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 711:
-    show(graph,xmin=plot_min,xmax=plot_max)
+    show(graph,xmin=plot_min,xmax=plot_max,aspect_ratio="auto")
-Line 685:
+Line 719:
-    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)))
+    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)))
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+Line 728:
-== Vector Calculus, 2-D Motion ==
+== Vector Calculus, 2-D Motion FIXME ==
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+Line 733:
-{{{
+{{{#!sagecell
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+Line 845:
-{{{
+{{{#!sagecell
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+Line 882:
-velocity = derivative( position(t) )
acceleration = derivative(velocity(t))
+velocity = derivative( position(t), t)
acceleration = derivative(velocity(t), t)
-Line 850:
+Line 885:
-speed_deriv = derivative(speed)
+speed_deriv = derivative(speed, t)
-Line 852:
+Line 887:
-dT = derivative(tangent(t))
+dT = derivative(tangent(t), t)
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+Line 890:
-## dB = derivative(binormal(t))
+## dB = derivative(binormal(t), t)
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+Line 968:
+== Multivariate Limits by Definition FIXME ==
by John Travis

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

{{{#!sagecell
##  An interactive way to demonstrate limits of multivariate functions of the form z = f(x,y)
##
##  John Travis
##  Mississippi College
##  
##  Spring 2011
##

#  Starting point for radius values before collapsing in as R approaches 0.
#  Functions ought to be "reasonable" within a circular domain of radius R surrounding 
#  the desired (x_0,y_0).
var('x,y,z')
Rmin=1/10
Rmax=2
@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'

#   converting f to cylindrical coordinates.  
    g(r,t) = f(x=r*cos(t)+x0,y=r*sin(t)+y0)

#   Sage graphing transformation used to see the original surface.
    cylinder = (r*cos(t)+x0,r*sin(t)+y0,z)
    surface = plot3d(g,(t,0,2*pi),(r,1/100,Rmax),transformation=cylinder,opacity=0.2)
    
#   Regraph the surface for smaller and smaller radii controlled by the slider.
    collapsing_surface = plot3d(g,(t,0,2*pi),(r,1/100,R),transformation=cylinder,rgbcolor=(0,1,0))
    
    G = surface+collapsing_surface
    html('Enter $(x_0 ,y_0 )$ above and see what happens as $ R \\rightarrow 0 $.')
    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 
 
    
        html('The red curves represent a couple of trajectories on the surface.  If they do not meet, then')
        html('there is also no limit.  (If computer hangs up, likely the computer can not do these limits.)')
        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)
}}}
{{attachment:3D_Limit_Defn.png}}


{{{#!sagecell
##  An interactive way to demonstrate limits of multivariate functions of the form z = f(x,y)
##  This one uses contour plots and so will work with functions that have asymptotic behavior.
##
##  John Travis
##  Mississippi College
##  
##  Spring 2011
##

#  An increasing number of contours for z = f(x,y) are utilized surrounding a desired (x_0,y_0).
#  A limit can be shown to exist at (x_0,y_0) provided the point stays trapped between adjacent 
#  contour lines as the number of lines increases.  If the contours change wildly near the point,
#  then a limit does not exist.
#  Looking for two different paths to approach (x_0,y_0) that utilize a different selection of colors
#  will help locate paths to use that exhibit the absence of a limit.

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.')    

    surface = contour_plot(f,(x,x0-1,x0+1),(y,y0-1,y0+1),cmap=True,colorbar=True,fill=False,contours=N)
    limit_point = point((x0,y0),color='red',size=30)
    show(limit_point+surface)}}}
{{attachment:3D_Limit_Defn_Contours.png}}
-Line 937:
+Line 1083:
-{{{
+{{{#!sagecell
-Line 990:
+Line 1136:
-{{{
%hide
%auto
+{{{#!sagecell
-Line 1051:
+Line 1195:
-{{{
+{{{#!sagecell
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+Line 1233:
-{{{
+{{{#!sagecell
-Line 1122:
+Line 1266:
+== Volumes over non-rectangular domains ==

by John Travis

http://www.sagenb.org/home/pub/2829/

{{{#!sagecell
##  Graphing surfaces over non-rectangular domains 
##  John Travis
##  Spring 2011
##
##
##  An updated version of this worksheet may be available at http://sagenb.mc.edu
##
##  Interact allows the user to input up to two inequality constraints on the
##  domain when dealing with functional surfaces
##
##  User inputs:
##    f = "top" surface with z = f(x,y)
##    g = "bottom" surface with z = g(x,y)
##    condition1 = a single boundary constraint.  It should not include && or | to join two conditions.
##    condition2 = another boundary constraint.  If there is only one constraint, just enter something true
##        or even just an x (or y) in the entry blank.
##
## 

var('x,y')

#  f is the top surface
#  g is the bottom surface
global f,g

#  condition1 and condition2 are the inequality constraints.  It would be nice
#  to have any number of conditions connected by $$ or | 
global condition1,condition2

@interact
def _(f=input_box(default=(1/3)*x^2 + (1/4)*y^2 + 5,label='$f(x)=$'),
        g=input_box(default=-1*x+0*y,label='$g(x)=$'),
        condition1=input_box(default= x^2+y^2<8,label='$Constraint_1=$'),
        condition2=input_box(default=y<sin(3*x),label='$Constraint_2=$'),
        show_3d=('Stereographic',false), show_vol=('Shade volume',true), 
        dospin = ('Spin?',true), 
        clr = color_selector('#faff00', label='Volume Color', widget='colorpicker', hide_box=True), 
        xx = range_slider(-5, 5, 1, default=(-3,3), label='X Range'),
        yy = range_slider(-5, 5, 1, default=(-3,3), label='Y Range'),
        auto_update=false):
    
    #  This is the top function actually graphed by using NaN outside domain
    def F(x,y):
        if condition1(x=x,y=y):
            if condition2(x=x,y=y):
                return f(x=x,y=y)
            else:
                return -NaN
        else:
            return -NaN

    # This is the bottom function actually graphed by using NaN outside domain
    def G(x,y):
        if condition1(x=x,y=y):
            if condition2(x=x,y=y):
                return g(x=x,y=y)
            else:
                return -NaN
        else:
            return -NaN
        
    P = Graphics()      
      
#  The graph of the top and bottom surfaces
    P_list = []
    P_list.append(plot3d(F,(x,xx[0],xx[1]),(y,yy[0],yy[1]),color='blue',opacity=0.9))
    P_list.append(plot3d(G,(x,xx[0],xx[1]),(y,yy[0],yy[1]),color='gray',opacity=0.9))
    
#  Interpolate "layers" between the top and bottom if desired

    if show_vol:
        ratios = range(10)

        def H(x,y,r):
            return (1-r)*F(x=x,y=y)+r*G(x=x,y=y)
        P_list.extend([
        plot3d(lambda x,y: H(x,y,ratios[1]/10),(x,xx[0],xx[1]),(y,yy[0],yy[1]),opacity=0.2,color=clr),
        plot3d(lambda x,y: H(x,y,ratios[2]/10),(x,xx[0],xx[1]),(y,yy[0],yy[1]),opacity=0.2,color=clr),
        plot3d(lambda x,y: H(x,y,ratios[3]/10),(x,xx[0],xx[1]),(y,yy[0],yy[1]),opacity=0.2,color=clr),
        plot3d(lambda x,y: H(x,y,ratios[4]/10),(x,xx[0],xx[1]),(y,yy[0],yy[1]),opacity=0.2,color=clr),
        plot3d(lambda x,y: H(x,y,ratios[5]/10),(x,xx[0],xx[1]),(y,yy[0],yy[1]),opacity=0.2,color=clr),
        plot3d(lambda x,y: H(x,y,ratios[6]/10),(x,xx[0],xx[1]),(y,yy[0],yy[1]),opacity=0.2,color=clr),
        plot3d(lambda x,y: H(x,y,ratios[7]/10),(x,xx[0],xx[1]),(y,yy[0],yy[1]),opacity=0.2,color=clr),
        plot3d(lambda x,y: H(x,y,ratios[8]/10),(x,xx[0],xx[1]),(y,yy[0],yy[1]),opacity=0.2,color=clr),
        plot3d(lambda x,y: H(x,y,ratios[9]/10),(x,xx[0],xx[1]),(y,yy[0],yy[1]),opacity=0.2,color=clr)
        ])
#            P = plot3d(lambda x,y: H(x,y,ratio/10),(x,xx[0],xx[1]),(y,yy[0],yy[1]),opacity=0.1)
             
           
#  Now, accumulate all of the graphs into one grouped graph.        
    P = sum(P_list[i] for i in range(len(P_list)))


    if show_3d:
        show(P,frame=true,axes=false,xmin=xx[0],xmax=xx[1],ymin=yy[0],ymax=yy[1],stereo='redcyan',figsize=(6,9),viewer='jmol',spin=dospin)
    else:
        show(P,frame=true,axes=false,xmin=xx[0],xmax=xx[1],ymin=yy[0],ymax=yy[1],figsize=(6,9),viewer='jmol',spin=dospin)
}}}
{{attachment:3D_Irregular_Volume.png}}

== Lateral Surface Area ==

by John Travis

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

{{{#!sagecell
##  Display and compute the area of the lateral surface between two surfaces
##  corresponding to the (scalar) line integral
##  John Travis
##  Spring 2011
##

var('x,y,t,s')
@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),
        xx = range_slider(-5, 5, 1, default=(-1,1), label='x view'),
        yy = range_slider(-5, 5, 1, default=(-2,2), label='y view'),
        in_3d = checkbox(default=true,label='3D'),
        smoother=checkbox(default=false),
        auto_update=true):
        
    ds = sqrt(derivative(u,t)^2+derivative(v,t)^2)
    
#   Set up the integrand to compute the line integral, making all attempts
#   to simplify the result so that it looks as nice as possible.    
    A = (f(x=u,y=v)-g(x=u,y=v))*ds.simplify_trig().simplify()
    
#   It is not expected that Sage can actually perform the line integral calculation.
#   So, the result displayed may not be a numerical value as expected.
#   Creating a good but harder example that "works" is desirable.
#   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_integral_approx = numerical_integral(A,a,b)[0]

    html(r'<font align=center size=+1>Lateral Surface $ \approx $ %s</font>'%str(line_integral_approx))

#   Plot the top function z = f(x,y) that is being integrated.
    G = plot3d(f,(x,xx[0],xx[1]),(y,yy[0],yy[1]),opacity=0.2)
    G += plot3d(g,(x,xx[0],xx[1]),(y,yy[0],yy[1]),opacity=0.2)

#   Add space curves on the surfaces "above" the domain curve (u(t),v(t)) 
    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')
    k=0
    if smoother:
        delw = 0.025
        lat_thick = 3
    else:
        delw = 0.10
        lat_thick = 10
    for w in (a,a+delw,..,b):
        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)
}}}
{{attachment:Lateral_Surface.png}}
-Line 1124:
+Line 1448:
-{{{
+{{{#!sagecell
-Line 1138:
+Line 1462:
+== Line Integrals in 3D Vector Field ==

by John Travis

http://www.sagenb.org/home/pub/2827/

{{{#!sagecell
##  This worksheet interactively computes and displays the line integral of a 3D vector field 
##  over a given smooth curve C
##  
##  John Travis
##  Mississippi College
##  06/16/11
##
##  An updated version of this worksheet may be available at http://sagenb.mc.edu
##

var('x,y,z,t,s')

@interact
def _(M=input_box(default=x*y*z,label="$M(x,y,z)$"),
        N=input_box(default=-y*z,label="$N(x,y,z)$"),
        P=input_box(default=z*y,label="$P(x,y,z)$"),
        u=input_box(default=cos(t),label="$x=u(t)$"),
        v=input_box(default=2*sin(t),label="$y=v(t)$"),
        w=input_box(default=t*(t-2*pi)/pi,label="$z=w(t)$"),
        tt = range_slider(-2*pi, 2*pi, pi/6, default=(0,2*pi), label='t Range'),
        xx = range_slider(-5, 5, 1, default=(-1,1), label='x Range'),
        yy = range_slider(-5, 5, 1, default=(-2,2), label='y Range'),
        zz = range_slider(-5, 5, 1, default=(-3,1), label='z Range'),
        in_3d=checkbox(true)):

#   setup the parts and then compute the line integral
    dr = [derivative(u(t),t),derivative(v(t),t),derivative(w(t),t)]
    A = (M(x=u(t),y=v(t),z=w(t))*dr[0]
        +N(x=u(t),y=v(t),z=w(t))*dr[1]
        +P(x=u(t),y=v(t),z=w(t))*dr[2])
    global line_integral
    line_integral = integral(A(t=t),t,tt[0],tt[1])
    
    html(r'<h2 align=center>$ \int_{C} \left \langle M,N,P \right \rangle dr $ = $ %s $ </h2>'%latex(line_integral))
    G = plot_vector_field3d((M,N,P),(x,xx[0],xx[1]),(y,yy[0],yy[1]),(z,zz[0],zz[1]),plot_points=6)
    G += parametric_plot3d([u,v,w],(t,tt[0],tt[1]),thickness='5',color='yellow')
    if in_3d:
        show(G,stereo='redcyan',spin=true)
    else:
        show(G,perspective_depth=true)
}}}
{{attachment:3D_Line_Integral.png}}

Diff for "interact/calculus"

Sage Interactions - Calculus

Root Finding Using Bisection

Newton's Method

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

A simple tangent line grapher

Numerical integrals with the midpoint rule

Numerical integrals with various rules

Some polar parametric curves

Function tool

Newton-Raphson Root Finding

Coordinate Transformations

Taylor Series

Illustration of the precise definition of a limit

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

Quadric Surface Plotter

The midpoint rule for numerically integrating a function of two variables

Gaussian (Legendre) quadrature

Vector Calculus, 2-D Motion FIXME

Vector Calculus, 3-D Motion

Multivariate Limits by Definition FIXME

Directional Derivatives

3D graph with points and curves

Approximating function in two variables by differential

Taylor approximations in two variables

Volumes over non-rectangular domains

Lateral Surface Area

Parametric surface example

Line Integrals in 3D Vector Field