Differences between revisions 40 and 57 (spanning 17 versions)

Sage Interactions - Calculus

goto interact main page

Contents

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://www.sagenb.org/home/pub/2828/

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 40 as of 2011-06-17 01:43:57 → 
  Size: 56871
  Editor: JohnTravis
  Comment:
+   ← Revision 57 as of 2013-04-24 18:45:37 → ⇥
  Size: 58970
  Editor: travis
  Comment:
-Deletions are marked like this.
+Additions are marked like this.
 Line 6:
-Line 9:
+Line 10:
-{{{
+{{{#!sagecell
-Line 58:
+Line 59:
-{{{
+{{{#!sagecell
-Line 100:
+Line 101:
-{{{
+{{{#!sagecell
-Line 116:
+Line 117:
-{{{
+{{{#!sagecell
-Line 136:
+Line 137:
-{{{
+{{{#!sagecell
-Line 164:
+Line 165:
-{{{
+{{{#!sagecell
-Line 237:
+Line 238:
-{{{
+{{{#!sagecell
-Line 250:
+Line 251:
-{{{
+{{{#!sagecell
-Line 348:
+Line 349:
-{{{
+{{{#!sagecell
-Line 454:
+Line 455:
-{{{
+{{{#!sagecell
-Line 456:
+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]])}}}
-Line 496:
+Line 521:
-{{{
+{{{#!sagecell
-Line 519:
+Line 544:
-{{{
+{{{#!sagecell
-Line 546:
+Line 571:
-{{{
+{{{#!sagecell
-Line 572:
+Line 597:
-{{{
+{{{#!sagecell
-Line 587:
+Line 612:
-{{{
+{{{#!sagecell
-Line 633:
+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)))
-Line 693:
+Line 728:
-== Vector Calculus, 2-D Motion ==
+== Vector Calculus, 2-D Motion FIXME ==
-Line 698:
+Line 733:
-{{{
+{{{#!sagecell
-Line 810:
+Line 845:
-{{{
+{{{#!sagecell
-Line 847:
+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)
-Line 855:
+Line 890:
-## dB = derivative(binormal(t))
+## dB = derivative(binormal(t), t)
-Line 933:
+Line 968:
-== Multivariate Limits by Definition ==
+== Multivariate Limits by Definition FIXME ==
-Line 936:
+Line 971:
-{{{
+http://www.sagenb.org/home/pub/2828/

{{{#!sagecell
-Line 944:
+Line 981:
+##  An updated version of this worksheet may be available at http://sagenb.mc.edu
-Line 970:
+Line 1008:
-{{{
+{{{#!sagecell
-Line 1007:
+Line 1045:
-{{{
+{{{#!sagecell
-Line 1060:
+Line 1098:
-{{{
%hide
%auto
+{{{#!sagecell
-Line 1121:
+Line 1157:
-{{{
+{{{#!sagecell
-Line 1159:
+Line 1195:
-{{{
+{{{#!sagecell
-Line 1197:
+Line 1233:
-{{{
+http://www.sagenb.org/home/pub/2829/

{{{#!sagecell
-Line 1201:
+Line 1239:
+##
##
##  An updated version of this worksheet may be available at http://sagenb.mc.edu
-Line 1299:
+Line 1340:
-{{{
+http://sagenb.mc.edu/home/pub/89/

{{{#!sagecell
-Line 1304:
+Line 1347:
+##
-Line 1306:
+Line 1350:
-@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 1361:
-        smoother=checkbox(default=false)):
+        in_3d = checkbox(default=true,label='3D'),
        smoother=checkbox(default=false),
        auto_update=true):
-Line 1317:
+Line 1365:
-    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 1369:
-    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 1374:
-    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 1380:
-    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))
+    html(r'<font align=center size=+1>Lateral Surface $ \approx $ %s</font>'%str(line_integral_approx))
-Line 1338:
+Line 1388:
-    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 1398:
-        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 1356:
+Line 1410:
-{{{
+{{{#!sagecell
-Line 1375:
+Line 1429:
-{{{
+http://www.sagenb.org/home/pub/2827/

{{{#!sagecell
-Line 1382:
+Line 1438:
+##
##  An updated version of this worksheet may be available at http://sagenb.mc.edu

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