The set of interactive pages with the following structure:
Example of Computations
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@interact
def _(N=(24,1,-1)):
var('x'); f=sin(x)*exp(1)^(-x)
p=plot(f,-1,6,thickness=5,color='#3636ff',alpha=.5)+\
sum([plot(f.taylor(x,0,i),-1,6,hue=cos(i/24),thickness=1,
legend_label=str(i)) for i in [1..N]])
ti=r'$Taylor \ Series \ f=sin \ x \cdot e^{-x}$'
p.show(ymin=-.5,ymax=.5,figsize=(7,5),
gridlines=True,title=ti,fontsize=12)
Type your own SageMath code lines below and click the
xxxxxxxxxxMd=Manifold(2,'Md')U=Md.open_subset('U'); V=Md.open_subset('V')XU.<x,y>=U.chart(); XV.<xp,yp>=V.chart('xp:x’ yp:y’')Md.declare_union(U,V)XU_to_XV=XU.transition_map( XV,(2*x/(x^2+y^2),2*y/(x^2+y^2)), intersection_name='W', restrictions1=x^2+y^2!=0, restrictions2=xp^2+yp^2!=0)R3=Manifold(3,'R^3',r'\mathbb{R}^3')XR3.<X,Y,Z>=R3.chart()Delta1=Md.diff_map( R3,{(XU,XR3):[2*x/(1+x^2+y^2),2*y/(1+x^2+y^2), (x^2+y^2-1)/(1+x^2+y^2)], (XV,XR3):[3*xp/(1+xp^2+yp^2),3*yp/(1+xp^2+yp^2), (1-xp^2-yp^2)/(1+xp^2+yp^2)]}, name='Delta1',latex_name=r'\Delta_1')Delta2=Md.diff_map( R3,{(XU, XR3):[x/(1+x^2+y^2),y/(1+x^2+y^2), (x^2+y^2-1)/(1+x^2+y^2)], (XV, XR3):[4*xp/(1+xp^2+yp^2),4*yp/(1+xp^2+yp^2), (1-xp^2-yp^2)/(1+xp^2+yp^2)]}, name='Delta2',latex_name=r'\Delta_2')cols=['darkblue','#ff3636','darkred','#3636ff']elements=[(XU,Delta1),(XV,Delta1),(XU,Delta2),(XV,Delta2)]p=sum([elements[i][0].plot( chart=XR3,mapping=elements[i][1], number_values=50,color=cols[i],label_axes=False) for i in range(4)])p.show(frame=False)This code cell was evaluated automatically.
xxxxxxxxxxdef _(a=(2,4,1),b=(4,8,1),c=(4,8,1),n=(2,8,1),m=(2,8,1)): var('t'); r=cos(a*t)^n+sin(b*t)^m+1/c; col='#3636ff' string='<center><font color=%s>$r = %s$</font></center>' pretty_print(html(string%(col,latex(r)))) p=parametric_plot((sin(t)*r,cos(t)*r),(0,2*pi), color=col,fill=True,fillcolor=col) p+=plot(r,(0,2*pi),linestyle='--',color=col,gridlines=True) show(p,figsize=(7,7), xmin=-2.2-1/c,xmax=6.4+1/c,ymin=-2-1/c,ymax=2+1/c)
xxxxxxxxxxg=Graphics()f1(x,y)=x^2+x*y+y^2+2*x+6*y+3f2(x,y)=x^2+x*y-y^2+2*x+6*y+3g+=implicit_plot(f1,(-8,8),(-8,8),color='#3636ff',linewidth=2,linestyle='-.')g+=text('$x^2+xy+y^2+2x+6y+3=0$',(1,-7),color='#3636ff',fontsize=14)g+=implicit_plot(f2,(-8,8),(-8,8),color='#ff36ff',linewidth=4,linestyle=':')g+=text('$x^2+xy-y^2+2x+6y+3=0$',(1,3),color='#ff36ff',fontsize=14)g.show(figsize=[7,7],gridlines=True)xxxxxxxxxximport numpy as np,pylab as pl from mpl_toolkits.mplot3d import Axes3Dx,y,z=np.indices((13,13,13))link=(x-6)**2+(y-5)**2+(z-6)**2==45f=pl.figure(figsize=(7,7))ax=f.add_subplot(111,projection='3d')ax.voxels(link,facecolors=pl.cm.bwr((x+y+x)*8), edgecolor='darkslategray')ti='$(x-6)^2+(y-5)^2+(z-6)^2=45$'pl.title(ti,fontsize=14,color='#3636ff')pl.tight_layout(); pl.show()xxxxxxxxxx%%html<p>Calculation the number of symbols in strings:</p><p id='script_output'>evaluate the next code cell</p>xxxxxxxxxx%%javascriptfunction getInteger(min,max) { return Math.floor(Math.random()*(max-min+1))+min;};var string='@@@***морозИсолнцеДЕНЬчудестный***@@@'+getInteger(1,19);document.getElementById('script_output') .innerHTML=string+' - '+string.length+' symbols';xxxxxxxxxx%%html<style>.text1 {color:darkred; font-size:200%; text-shadow:3px 3px 3px #aaa;}.text2 {color:darkslategray; font-size:200%; text-shadow:3px 3px 3px #bbb;}.text3 {color:steelblue; font-size:200%; text-shadow:3px 3px 3px #ccc;}</style><p class='text1'>📕 $\mathfrak {Choose \ your \ style!}$</p><p class='text2'>📓 $\mathbb {Choose \ your \ style!}$</p><p class='text3'>📘 $\mathscr {Choose \ your \ style!}$</p>