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== Multivariate Limits by Definition FIXME ==  == Multivariate Limits by Definition == 
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http://www.sagenb.org/home/pub/2828/  http://sagenb.mc.edu/home/pub/97/ 
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## An updated version of this worksheet may be available at http://sagenb.mc.edu  
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var('x,y,z') Rmin=1/10 

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@interact def _(f=input_box(default=(x^3y^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^2y^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' 
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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)) 
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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([x0t,y0t^(3/5),f(x=x0t,y=y0t^(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((x0Rmin,y0Rmin^(3/5),f(x=x0Rmin,y=y0Rmin^(3/5))),(x0R2,y0R2^(3/5),f(x=x0R2,y=y0R2^(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=x0t,y=y0t^(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) 
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
 NewtonRaphson 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, 2D Motion FIXME
 Vector Calculus, 3D Motion
 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 nonrectangular domains
 Lateral Surface Area
 Parametric surface example
 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
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.
NewtonRaphson Root Finding
by Neal Holtz
This allows user to display the NewtonRaphson 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 restarted, 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, 2D Motion FIXME
By Rob Beezer
A fast_float() version is available in a worksheet
Vector Calculus, 3D Motion
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(x^{2} + y^{2}) cos(y) exp((x^{2}+y^{2})/2).
Volumes over nonrectangular 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/