Differences between revisions 54 and 56 (spanning 2 versions)
 ⇤ ← Revision 54 as of 2012-04-26 15:50:40 → Size: 57403 Editor: jason Comment: ← Revision 56 as of 2012-05-09 01:38:57 → ⇥ Size: 58534 Editor: jason Comment: Deletions are marked like this. Additions are marked like this. Line 457: 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=4square_length=.05 Line 460: 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), def trans(x=input_box(x, label="x",type=SR),         y=input_box(y, label="y",type=SR), Line 465: Line 474: u_range=input_box('[-5..5]', label="u lines"),         v_range=input_box('[-5..5]', 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     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)      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)     html("$$x=%s, \: y=%s$$"%(latex(x), latex(y)))     html.table([[uvplot,xyplot]])}}} 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 703: Line 728: == Vector Calculus, 2-D Motion == == Vector Calculus, 2-D Motion FIXME == Line 943: Line 968: == Multivariate Limits by Definition == == Multivariate Limits by Definition FIXME == Line 1311: Line 1336: == Lateral Surface Area == == Lateral Surface Area FIXME ==

# Sage Interactions - Calculus

by William Stein

## Newton's Method

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

by William Stein

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.

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

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

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

by John Travis

## 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

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).

by John Travis

by John Travis

## Parametric surface example

by Marshall Hampton

## Line Integrals in 3D Vector Field

by John Travis

interact/calculus (last edited 2020-08-11 14:10:09 by kcrisman)