|
Size: 58192
Comment:
|
Size: 62750
Comment:
|
| Deletions are marked like this. | Additions are marked like this. |
| Line 10: | Line 10: |
| [[http://aleph.sagemath.org/?z=eJytU02P2yAQvedXjCxFC4njJP06bOOqrdqe9hBVbS-rNMIY1ki2sQDvev99B5zYzq62l5aDPcDw5jHvkQsJmbKCu2MlXKFzImNgMWQxiMbS6xngcOaxD_yQkIJMjpJZd5SlZu5IZHLPjGJZKSyht5sDDcmi46Jx8Mk5o7LWia_GaDPiNMzaMFG1E-aelRaBbwmLM3roqz5oX8qXIK_oe89nmHtuIemhUKWAH6YVIzLHNMKWGV0jxMib-dOEIZLMQpj5kIeQ0yFPSWCZJZJT2Pma12CEa00NPB6pTrMlWyDKDjYjAz98ExEbf3xYF6XP5wsk8FI-Vskm-VZcZhmGYsEvVrZ9P2OIJFStdVCwewEMrLpDqgWr7wTSBVeIgTWQuY3nlkbz0OXxDudbJaxpRJ2TsE1nhatKEu2K7YcvukVxYW8EV1bpGr5r7eCbqnNV38FP67-fg4twc7fGExGdfQy4jLtZjiZDl_jbaUs6Civo0GWDmhva372fbWnwXkpWr2Oy2r5LktWW0pMVUauOprKPe0dsN78x6Fcag0Uh6jfmNpo_scullcGpSlzoiqf-8hpOVXtfjxpMAE_1p5LoptFWOdErw1yQBLvcaEy1oOWFRtGIZQv9QJpSu5EE16U26ZUR-RU2qatUnTL_Z1161vOJZU6EjNcrjaZeHLii-X2zDDZr-g7O-8qL6HW1AaAUNRnaNcneI8bLZMe8wj_OffKInIk3gg9VTShdA9kmbxeX-JODN3jQthUpVS3ILeFxsVDYBJKH4EBBagMKF3icU299UbeVpy4meHABuHyOuCrWb2gAwcnST_4Dcj5Fzv8FOXhiD0u4eSb_HxOSmYw%3D|Interact]] {{{ |
{{{#!sagecell |
| Line 30: | Line 29: |
| 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])): |
| Line 36: | Line 35: |
| time c, intervals = bisect_method(f, a, b, eps) | c, intervals = bisect_method(f, a, b, eps) |
| Line 58: | Line 57: |
| 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 |
| Line 78: | Line 77: |
| html("<h1>Double Precision Root Finding Using Newton's Method</h1>") | pretty_print(html("<h1>Double Precision Root Finding Using Newton's Method</h1>")) |
| Line 83: | Line 82: |
| time z, iterates = newton_method(f, c, eps) | z, iterates = newton_method(f, c, eps) |
| Line 88: | Line 87: |
| html(iterates) | pretty_print(html(iterates)) |
| Line 100: | Line 99: |
| http://sagenb.org/home/pub/2823/ {{{ |
https://cloud.sagemath.com/projects/19575ea0-317e-402b-be57-368d04c113db/files/pub/2801-2901/2823.sagews {{{#!sagecell |
| Line 118: | Line 117: |
| {{{ html('<h2>Tangent line grapher</h2>') |
{{{#!sagecell pretty_print(html('<h2>Tangent line grapher</h2>')) |
| Line 130: | Line 129: |
| 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] |
| Line 138: | Line 137: |
| {{{ | {{{#!sagecell #find_maximum_on_interval and find_minimum_on_interval are deprecated #use find_local_maximum find_local_minimum instead #see http://trac.sagemath.org/2607 for details -RRubalcaba |
| Line 153: | Line 156: |
| 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('$\int_{a}^{b}{f(x) dx} {\\approx} \sum_i{f(x_i) \Delta x}$') |
min_y = min(0, sage.numerical.optimize.find_local_minimum(func,a,b)[0]) max_y = max(0, sage.numerical.optimize.find_local_maximum(func,a,b)[0]) pretty_print(html('<h3>Numerical integrals with the midpoint rule</h3>')) pretty_print(html('$\int_{a}^{b}{f(x) dx} {\\approx} \sum_i{f(x_i) \Delta x}$')) |
| Line 166: | Line 169: |
| {{{ | {{{#!sagecell |
| Line 168: | Line 171: |
| #find_maximum_on_interval and find_minimum_on_interval are deprecated #use find_local_maximum find_local_minimum instead #see http://trac.sagemath.org/2607 for details -RRubalcaba |
|
| Line 191: | Line 197: |
| 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] |
| Line 194: | Line 200: |
| 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] |
| Line 205: | Line 211: |
| 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]) |
| Line 222: | Line 228: |
| html(r''' | pretty_print(html(r''' |
| Line 232: | Line 238: |
| ''' % (numerical_answer, number_of_subdivisions, sum_html, num_html, estimated_answer)) | ''' % (numerical_answer, number_of_subdivisions, sum_html, num_html, estimated_answer))) |
| Line 239: | Line 245: |
| {{{ | {{{#!sagecell |
| Line 252: | Line 258: |
| {{{ | {{{#!sagecell |
| Line 350: | Line 356: |
| {{{ | {{{#!sagecell |
| Line 452: | Line 458: |
| == Coordinate Transformations == | == Coordinate Transformations (FIXME in Jupyter) == |
| Line 456: | Line 462: |
| {{{ | {{{#!sagecell |
| Line 458: | Line 464: |
| # 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 461: | Line 476: |
| 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 466: | Line 480: |
| 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()) pretty_print(table([[uvplot,xyplot]])) |
| Line 498: | Line 529: |
| {{{ | {{{#!sagecell |
| Line 521: | Line 552: |
| {{{ | {{{#!sagecell |
| Line 548: | Line 579: |
| {{{ | {{{#!sagecell |
| Line 574: | Line 605: |
| {{{ | {{{#!sagecell |
| Line 589: | Line 620: |
| {{{ | {{{#!sagecell |
| Line 608: | Line 639: |
| sin,cos = math.sin,math.cos | |
| Line 635: | Line 666: |
| {{{ from scipy.special.orthogonal import p_roots |
{{{#!sagecell import scipy import numpy from scipy.special.orthogonal import p_roots, t_roots, u_roots |
| Line 645: | Line 678: |
| '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 652: | Line 688: |
| 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 656: | Line 693: |
| 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 665: | Line 703: |
| scaled_ff = fast_float(scaled_func) | scaled_ff = fast_float(scaled_func, 'x') |
| Line 673: | Line 711: |
| 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 679: | Line 719: |
| show(graph,xmin=plot_min,xmax=plot_max) | show(graph,xmin=plot_min,xmax=plot_max,aspect_ratio="auto") |
| Line 687: | Line 727: |
| 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 700: | Line 741: |
| {{{ | {{{#!sagecell |
| Line 731: | Line 772: |
| velocity = derivative( position(t) ) acceleration = derivative(velocity(t)) |
velocity = derivative(position(t), t) acceleration = derivative(velocity(t), t) |
| Line 734: | Line 775: |
| speed_deriv = derivative(speed) | speed_deriv = derivative(speed, t) |
| Line 736: | Line 777: |
| dT = derivative(tangent(t)) | dT = derivative(tangent(t), t) |
| Line 807: | Line 848: |
| == Vector Calculus, 3-D Motion == | == Vector Calculus, 3-D Motion (FIXME) == |
| Line 812: | Line 853: |
| {{{ | {{{#!sagecell |
| Line 849: | Line 890: |
| velocity = derivative( position(t) ) acceleration = derivative(velocity(t)) |
velocity = derivative( position(t), t) acceleration = derivative(velocity(t), t) |
| Line 852: | Line 893: |
| speed_deriv = derivative(speed) | speed_deriv = derivative(speed, t) |
| Line 854: | Line 895: |
| dT = derivative(tangent(t)) | dT = derivative(tangent(t), t) |
| Line 857: | Line 898: |
| ## dB = derivative(binormal(t)) | ## dB = derivative(binormal(t), t) |
| Line 938: | Line 979: |
| http://www.sagenb.org/home/pub/2828/ {{{ |
http://sagenb.mc.edu/home/pub/97/ {{{#!sagecell |
| Line 948: | Line 989: |
| ## An updated version of this worksheet may be available at http://sagenb.mc.edu | |
| Line 953: | Line 993: |
| var('x,y,z') Rmin=1/10 |
|
| Line 955: | Line 996: |
| @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 964: | Line 1016: |
| Line 966: | Line 1018: |
| 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 968: | Line 1020: |
| 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 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) |
| Line 975: | Line 1054: |
| {{{ | {{{#!sagecell |
| Line 992: | Line 1072: |
| 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='$y_0$')): html('Enter $(x_0 ,y_0 )$ above and see what happens as the number of contour levels $\\rightarrow \infty $.') html('A surface will have a limit in the center of this graph provided there is not a sudden change in color there.') # Need to make certain the min and max contour lines are not huge due to asymptotes. If so, clip and start contours at some reasonable # values so that there are a nice collection of contours to show around the desired point. |
| Line 1002: | Line 1088: |
| surface += parametric_plot([R*cos(u),R*sin(u)],[0,2*pi],color='black') # Nice to use if f=x*y^2/(x^2 + y^4) # var('u') # surface += parametric_plot([u^2,u],[u,-1,1],color='black') |
|
| Line 1003: | Line 1093: |
| show(limit_point+surface)}}} | # show(limit_point+surface) pretty_print(table([[surface],['hi']])) }}} |
| Line 1012: | Line 1104: |
| {{{ | {{{#!sagecell |
| Line 1065: | Line 1157: |
| {{{ %hide %auto |
{{{#!sagecell |
| Line 1126: | Line 1216: |
| {{{ | {{{#!sagecell |
| Line 1164: | Line 1254: |
| {{{ | {{{#!sagecell |
| Line 1202: | Line 1292: |
| http://www.sagenb.org/home/pub/2829/ {{{ |
https://cloud.sagemath.com/projects/19575ea0-317e-402b-be57-368d04c113db/files/pub/2801-2901/2829.sagews {{{#!sagecell |
| Line 1305: | Line 1395: |
| == Lateral Surface Area == | == Lateral Surface Area (FIXME in Jupyter) == |
| Line 1309: | Line 1399: |
| http://www.sagenb.org/home/pub/2826/ {{{ |
http://sagenb.mc.edu/home/pub/89/ {{{#!sagecell |
| Line 1316: | Line 1406: |
| ## | |
| Line 1318: | Line 1409: |
| @interact def _(f=input_box(default=6-4*x^2-y^2*2/5,label='$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 1327: | Line 1420: |
| smoother=checkbox(default=false)): | in_3d = checkbox(default=true,label='3D'), smoother=checkbox(default=false), auto_update=true): |
| Line 1329: | Line 1424: |
| ds = sqrt(derivative(u(t),t)^2+derivative(v(t),t)^2) | ds = sqrt(derivative(u,t)^2+derivative(v,t)^2) |
| Line 1333: | Line 1428: |
| 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 1338: | Line 1433: |
| 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 1340: | Line 1439: |
| 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 1350: | Line 1447: |
| 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 1360: | Line 1457: |
| 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 1366: | Line 1467: |
| == Parametric surface example == | == Parametric surface example (FIXME in Jupyter) == |
| Line 1368: | Line 1469: |
| {{{ | {{{#!sagecell |
| Line 1387: | Line 1488: |
| http://www.sagenb.org/home/pub/2827/ {{{ |
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 {{{#!sagecell |
Sage Interactions - Calculus
goto interact main page
Contents
-
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 (FIXME in Jupyter)
- 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
- Vector Calculus, 3-D Motion (FIXME)
- Multivariate Limits by Definition
- 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 (FIXME in Jupyter)
- Parametric surface example (FIXME in Jupyter)
- Line Integrals in 3D Vector Field
Root Finding Using Bisection
by William Stein
Newton's Method
Note that there is a more complicated Newton's method below.
by William Stein
A contour map and 3d plot of two inverse distance functions
by William Stein
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 (FIXME in Jupyter)
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
By Rob Beezer
A fast_float() version is available in a worksheet
Vector Calculus, 3-D Motion (FIXME)
by Rob Beezer
Available as a worksheet
Multivariate Limits by Definition
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).
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
Lateral Surface Area (FIXME in Jupyter)
by John Travis
http://sagenb.mc.edu/home/pub/89/
Parametric surface example (FIXME in Jupyter)
by Marshall Hampton
Line Integrals in 3D Vector Field
by John Travis
