Differences between revisions 88 and 117 (spanning 29 versions)
⇤ ← Revision 88 as of 2017-03-18 15:01:26 →
Size: 63134
Editor: mforets
Comment: fix DeprecationWarning and an AttributeError in Numerical integrals with the midpoint rule
← Revision 117 as of 2020-08-11 14:10:09 → ⇥
Size: 63144
Editor: kcrisman
Comment:
Deletions are marked like this. Additions are marked like this.
Line 27: Line 27:
            raise ValueError, "f must have a sign change in the interval (%s,%s)"%(a,b)             raise ValueError("f must have a sign change in the interval (%s,%s)"%(a,b))
Line 33: Line 33:
     print "eps = %s"%float(eps)      print("eps = %s" % float(eps))
Line 37: Line 37:
         print "f must have opposite sign at the endpoints of the interval"          print("f must have opposite sign at the endpoints of the interval")
Line 40: Line 40:
         print "root =", c
         print "f(c) = %r"%f(x=c)
         print "iterations =", len(intervals)		          print("root =", c)
         print("f(c) = %r" % f(x=c))
         print("iterations =", len(intervals))
Line 69: Line 69:
    for i in xrange(maxiter):     for i in range(maxiter):
Line 81: Line 81:
     print "eps = %s"%float(eps)      print("eps = %s"%float(eps))
Line 83: Line 83:
     print "root =", z
     print "f(c) = %r"%f(x=z)		      print("root = {}".format(z))
     print("f(c) = %r" % f(x=z))
Line 86: Line 86:
     print "iterations =", n      print("iterations = {}".format(n))
Line 127: Line 127:
    print 'Tangent line is y = ' + tanf._repr_()     print('Tangent line is y = ' + tanf._repr_())
Line 155: Line 155:
    pretty_print(html('$\int_{a}^{b}{f(x) dx} {\\approx} \sum_i{f(x_i) \Delta x}$'))
    print "\n\nSage numerical answer: " + str(integral_numerical(func,a,b,max_points = 200)[0])
    print "Midpoint estimated answer: " + str(RDF(dx*sum([midys[q] for q in range(n)])))		     pretty_print(html(r'$\int_{a}^{b}{f(x) dx} {\approx} \sum_i{f(x_i) \Delta x}$'))
    print("\n\nSage numerical answer: " + str(integral_numerical(func,a,b,max_points = 200)[0]))
    print("Midpoint estimated answer: " + str(RDF(dx*sum([midys[q] for q in range(n)]))))
Line 166: Line 166:
# by Nick Alexander (based on the work of Marshall Hampton)
#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 179: Line 174:
    t = sage.calculus.calculus.var('t')     t = var('t')
Line 210: Line 205:
    # html('<h3>Numerical integrals with the midpoint rule</h3>')     pretty_print(html('<h3>Numerical integral with the {} rule</h3>'.format(endpoint_rule)))
Line 218: Line 213:
    sum_html = "%s \cdot \\left[ %s \\right]" % (dx, ' + '.join([ "f(%s)" % cap(i) for i in xs ]))
    num_html = "%s \cdot \\left[ %s \\right]" % (dx, ' + '.join([ str(cap(i)) for i in ys ]))		     sum_html = "%s \\cdot \\left[ %s \\right]" % (dx, ' + '.join([ "f(%s)" % cap(i) for i in xs ]))
    num_html = "%s \\cdot \\left[ %s \\right]" % (dx, ' + '.join([ str(cap(i)) for i in ys ]))
Line 224: Line 219:
    html(r'''
    <div class="math">
    \begin{align*}
   \int_{a}^{b} {f(x) \, dx} & = %s \\\
   \sum_{i=1}^{%s} {f(x_i) \, \Delta x}
      & = %s \\\
   & = %s \\\
   & = %s .
    \end{align*}
    </div>
    ''' % (numerical_answer, number_of_subdivisions, sum_html, num_html, estimated_answer))		     pretty_print(html(r'''
    <div class="math"> 
    \begin{align*}   \int_{a}^{b} {f(x) \, dx} & = %s \\\   \sum_{i=1}^{%s} {f(x_i) \, \Delta x} & = %s \\\   & = %s \\\   & = %s . \end{align*} </div>'''
                      % (numerical_answer, number_of_subdivisions, sum_html, num_html, estimated_answer)))
Line 265: Line 257:
    except TypeError, msg:
        print msg[-200:]
        print "Unable to make sense of f,g, or a as symbolic expressions."		     except TypeError as msg:
        print(msg[-200:])
        print("Unable to make sense of f,g, or a as symbolic expressions.")
Line 382: Line 374:
                     vertical_alignment="bottom" if f(x0) < 0 else "top" )                      vertical_alignment="bottom" if f(x=x0) < 0 else "top" )
Line 398: Line 390:
        fi = RR(f(xi))
        fpi = RR(df(xi))		         fi = RR(f(x=xi))
        fpi = RR(df(x=xi))
Line 434: Line 426:
                             vertical_alignment="bottom" if f(xip1) < 0 else "top" )                              vertical_alignment="bottom" if f(x=xip1) < 0 else "top" )
Line 450: Line 442:
            html( t )             pretty_print(html( t ))
Line 454: Line 446:
== Coordinate Transformations (FIXME in Jupyter) == == Coordinate Transformations ==
Line 474: Line 466:
         u_percent=slider(0,1,0.05,label="<font color='red'>u</font>", default=.7),
         v_percent=slider(0,1,0.05,label="<font color='blue'>v</font>", default=.7),		          u_percent=slider(0,1,0.05,label="u", default=.7),
         v_percent=slider(0,1,0.05,label="v", default=.7),
Line 505: Line 497:
    jacobian=abs(T.diff().det()).simplify_full()     jacobian(u,v)=abs(T.diff().det()).simplify_full()
Line 516: Line 508:
    pretty_print(table([[uvplot,xyplot]]))     show(graphics_array([uvplot,xyplot]))
Line 530: Line 522:
dot = point((x0,f(x0)),pointsize=80,rgbcolor=(1,0,0))
@interact
def _(order=(1..12)):		 dot = point((x0,f(x=x0)),pointsize=80,rgbcolor=(1,0,0))
@interact
def _(order=[1..12]):
Line 535: Line 527:
    pretty_print(html('$f(x)\;=\;%s$'%latex(f)))
    pretty_print(html('$\hat{f}(x;%s)\;=\;%s+\mathcal{O}(x^{%s})$'%(x0,latex(ft),order+1)))		     pretty_print(html(r'$f(x)\;=\;%s$'%latex(f)))
    pretty_print(html(r'$\hat{f}(x;%s)\;=\;%s+\mathcal{O}(x^{%s})$'%(x0,latex(ft),order+1)))
Line 557: Line 549:
def delta_epsilon(f = input_box(default=(x^2-x)/(x-1)), a=input_box(default=1), L = input_box(default=1), delta=input_box(label="δ",default=0.1), epsilon=input_box(label="ε",default=0.1), xm=input_box(label="<i>x</i><sub>min</sub>",default=-1), xM=input_box(label="<i>x</i><sub>max</sub>",default=4)): def delta_epsilon(f = input_box(default=(x^2-x)/(x-1), label="$f$"), a=input_box(default=1, label="$a$"), L = input_box(default=1, label="$L$"), delta=input_box(label=r"$\delta$",default=0.1), epsilon=input_box(label=r"$\varepsilon$",default=0.1), xm=input_box(label=r"$x_{min}$",default=-1), xM=input_box(label=r"$x_{max}$",default=4)):
Line 579: Line 571:
    pretty_print(html('<h3>A graphical illustration of $\lim_{x -> 0} \sin(x)/x =1$</h3>'))
    pretty_print(html('Below is the unit circle, so the length of the <font color=red>red line</font> is |sin(x)|'))
    pretty_print(html('and the length of the <font color=blue>blue line</font> is |tan(x)| where x is the length of the arc.'))
    pretty_print(html('From the picture, we see that |sin(x)| $\le$ |x| $\le$ |tan(x)|.'))
    pretty_print(html('It follows easily from this that cos(x) $\le$ sin(x)/x $\le$ 1 when x is near 0.'))
    pretty_print(html('As $\lim_{x ->0} \cos(x) =1$, we conclude that $\lim_{x -> 0} \sin(x)/x =1$.'))		     pretty_print(html(r'<h3>A graphical illustration of $\lim_{x -> 0} \sin(x)/x =1$</h3>'))
    pretty_print(html(r'Below is the unit circle, so the length of the <font color=red>red line</font> is |sin(x)|'))
    pretty_print(html(r'and the length of the <font color=blue>blue line</font> is |tan(x)| where x is the length of the arc.'))
    pretty_print(html(r'From the picture, we see that |sin(x)| $\le$ |x| $\le$ |tan(x)|.'))
    pretty_print(html(r'It follows easily from this that cos(x) $\le$ sin(x)/x $\le$ 1 when x is near 0.'))
    pretty_print(html(r'As $\lim_{x ->0} \cos(x) =1$, we conclude that $\lim_{x -> 0} \sin(x)/x =1$.'))
Line 605: Line 597:
def quads(q = selector(quadrics.keys()), a = slider(0,5,1/2,default = 1)): def quads(q = selector(list(quadrics)), a = slider(0,5,1/2,default = 1)):
Line 607: Line 599:
    if a==0 or q=='Cone': html('<center>$'+latex(f)+' \ $'+ '(degenerate)</center>')
    else: html('<center>$'+latex(f)+'$ </center>')		     if a==0 or q=='Cone': pretty_print(latex(f), "   (degenerate)")
    else: pretty_print(latex(f))
Line 636: Line 628:
pretty_print(html("<h1>The midpoint rule for a function of two variables</h1>")) pretty_print(html(r"<h1>The midpoint rule for a function of two variables</h1>"))
Line 651: Line 643:
    pretty_print(html("$$\int_{"+str(R16(y_start))+"}^{"+str(R16(y_end))+"} "+ "\int_{"+str(R16(x_start))+"}^{"+str(R16(x_end))+"} "+func+"\ dx \ dy$$"))
    pretty_print(html('<p style="text-align: center;">Numerical approximation: ' + str(num_approx)+'</p>'))		     pretty_print(html(r"$\int_{"+str(R16(y_start))+r"}^{"+str(R16(y_end))+r"} "+ r"\int_{"+str(R16(x_start))+r"}^{"+str(R16(x_end))+r"} "+latex(SR(func))+r"\ dx \ dy$"))
    pretty_print(html(r'<p style="text-align: center;">Numerical approximation: ' + str(num_approx)+r'</p>'))
Line 668: Line 660:
from numpy import linspace from numpy import linspace, asanyarray, diff
Line 720: Line 712:
    y_val = map(scaled_ff,x_val)     y_val = [*map(scaled_ff,x_val)]
Line 723: Line 715:
    pretty_print(html("$$\sum_{i=1}^{i=%s}w_i\left(%s\\right)= %s\\approx %s =\int_{-1}^{1}%s \,dx$$"%(n,     pretty_print(html(r"$$\sum_{i=1}^{i=%s}w_i\left(%s\right)= %s\approx %s =\int_{-1}^{1}%s \,dx$$"%(n,
Line 726: Line 718:
    print "Trapezoid: %s, Simpson: %s, \nMethod: %s, Real: %s"%tuple(error_data)     print("Trapezoid: %s, Simpson: %s, \nMethod: %s, Real: %s" % tuple(error_data))
Line 764: Line 756:
path = parametric_plot( position(t).list(), (t, start, stop), color = "black" ) path = parametric_plot( position.list(), (t, start, stop), color = "black" )
Line 768: Line 760:
velocity = derivative(position(t), t)
acceleration = derivative(velocity(t), t)		 velocity = derivative(position, t)
acceleration = derivative(velocity, t)
Line 773: Line 765:
dT = derivative(tangent(t), t) dT = derivative(tangent, t)
Line 794: Line 786:
    pos_tzero = position(t0)     pos_tzero = position(t=t0)
Line 798: Line 790:
    speed_component = speed(t0)
    tangent_component = speed_deriv(t0)
    normal_component = sqrt( acceleration(t0).norm()^2 - tangent_component^2 )		     speed_component = speed(t=t0)
    tangent_component = speed_deriv(t=t0)
    normal_component = sqrt( acceleration(t=t0).norm()^2 - tangent_component^2 )
Line 806: Line 798:
    tan = arrow(pos_tzero, pos_tzero + tangent(t0), rgbcolor=(0,1,0) )
    vel = arrow(pos_tzero, pos_tzero + velocity(t0), rgbcolor=(0,0.5,0))
    nor = arrow(pos_tzero, pos_tzero + normal(t0), rgbcolor=(0.5,0,0))
    acc = arrow(pos_tzero, pos_tzero + acceleration(t0), rgbcolor=(1,0,1))
    tancomp = arrow(pos_tzero, pos_tzero + tangent_component*tangent(t0), rgbcolor=(1,0,1) )
    norcomp = arrow(pos_tzero, pos_tzero + normal_component*normal(t0), rgbcolor=(1,0,1))		     tan = arrow(pos_tzero, pos_tzero + tangent(t=t0), rgbcolor=(0,1,0) )
    vel = arrow(pos_tzero, pos_tzero + velocity(t=t0), rgbcolor=(0,0.5,0))
    nor = arrow(pos_tzero, pos_tzero + normal(t=t0), rgbcolor=(0.5,0,0))
    acc = arrow(pos_tzero, pos_tzero + acceleration(t=t0), rgbcolor=(1,0,1))
    tancomp = arrow(pos_tzero, pos_tzero + tangent_component*tangent(t=t0), rgbcolor=(1,0,1) )
    norcomp = arrow(pos_tzero, pos_tzero + normal_component*normal(t=t0), rgbcolor=(1,0,1))
Line 833: Line 825:
    print "Position vector defined as r(t)=", position(t)
    print "Speed is ", N(speed(t0))
    print "Curvature is ", N(curvature)		     print("Position vector defined as r(t)={}".format(position))
    print("Speed is {}".format(N(speed(t=t0))))
    print("Curvature is {}".format(N(curvature)))
Line 844: Line 836:
== Vector Calculus, 3-D Motion (FIXME) == == Vector Calculus, 3-D Motion ==
Line 865: Line 857:
assume(t, 'real')
Line 882: Line 875:
path = parametric_plot3d( position(t).list(), (t, start, stop), color = "black" ) path = parametric_plot3d( position.list(), (t, start, stop), color = "black" )
Line 886: Line 879:
velocity = derivative( position(t), t)
acceleration = derivative(velocity(t), t)		 velocity = derivative( position, t)
acceleration = derivative(velocity, t)
Line 891: Line 884:
dT = derivative(tangent(t), t) dT = derivative(tangent, t)
Line 894: Line 887:
## dB = derivative(binormal(t), t) ## dB = derivative(binormal, t)
Line 915: Line 908:
    pos_tzero = position(t0)     pos_tzero = position(t=t0)
Line 919: Line 912:
    speed_component = speed(t0)
    tangent_component = speed_deriv(t0)
    normal_component = sqrt( acceleration(t0).norm()^2 - tangent_component^2 )		     speed_component = speed(t=t0)
    tangent_component = speed_deriv(t=t0)
    normal_component = sqrt( acceleration(t=t0).norm()^2 - tangent_component^2 )
Line 928: Line 921:
    tan = arrow3d(pos_tzero, pos_tzero + tangent(t0), rgbcolor=(0,1,0) )
    vel = arrow3d(pos_tzero, pos_tzero + velocity(t0), rgbcolor=(0,0.5,0))
    nor = arrow3d(pos_tzero, pos_tzero + normal(t0), rgbcolor=(0.5,0,0))
    bin = arrow3d(pos_tzero, pos_tzero + binormal(t0), rgbcolor=(0,0,0.5))
    acc = arrow3d(pos_tzero, pos_tzero + acceleration(t0), rgbcolor=(1,0,1))
    tancomp = arrow3d(pos_tzero, pos_tzero + tangent_component*tangent(t0), rgbcolor=(1,0,1) )
    norcomp = arrow3d(pos_tzero, pos_tzero + normal_component*normal(t0), rgbcolor=(1,0,1))		     tan = arrow3d(pos_tzero, pos_tzero + tangent(t=t0), rgbcolor=(0,1,0) )
    vel = arrow3d(pos_tzero, pos_tzero + velocity(t=t0), rgbcolor=(0,0.5,0))
    nor = arrow3d(pos_tzero, pos_tzero + normal(t=t0), rgbcolor=(0.5,0,0))
    bin = arrow3d(pos_tzero, pos_tzero + binormal(t=t0), rgbcolor=(0,0,0.5))
    acc = arrow3d(pos_tzero, pos_tzero + acceleration(t=t0), rgbcolor=(1,0,1))
    tancomp = arrow3d(pos_tzero, pos_tzero + tangent_component*tangent(t=t0), rgbcolor=(1,0,1) )
    norcomp = arrow3d(pos_tzero, pos_tzero + normal_component*normal(t=t0), rgbcolor=(1,0,1))
Line 958: Line 951:
    print "Position vector: r(t)=", position(t)
    print "Speed is ", N(speed(t0))
    print "Curvature is ", N(curvature)
    ## print "Torsion is ", N(torsion)
    print
    print "Right-click on graphic to zoom to 400%"
    print "Drag graphic to rotate"		     print("Position vector: r(t)=", position)
    print("Speed is ", N(speed(t=t0)))
    print("Curvature is ", N(curvature))
    ## print("Torsion is ", N(torsion))
    print()
    print("Right-click on graphic to zoom to 400%")
    print("Drag graphic to rotate")
Line 1040: Line 1033:
        pretty_print(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)))         pretty_print(html(r'<center><font color="red">$\lim_{(x,?)\rightarrow(x_0,y_0)} f(x,y) =%s$</font>'%str(limit_x)+r' and <font color="red">$\lim_{(?,y)\rightarrow(x_0,y_0)} f(x,y) =%s$</font></center>'%str(limit_y)))
Line 1051: Line 1044:
Line 1077: Line 1069:
    pretty_print(html('Enter $(x_0 ,y_0 )$ above and see what happens as the number of contour levels $\\rightarrow \infty $.'))     pretty_print(html(r'Enter $(x_0 ,y_0 )$ above and see what happens as the number of contour levels $\rightarrow \infty $.'))
Line 1090: Line 1082:
    pretty_print(table([[surface],['hi']]))     show(surface)
Line 1259: Line 1251:
      order=(1..10)):       order=[1..10]):
Line 1391: Line 1383:
== Lateral Surface Area (FIXME in Jupyter) == == Lateral Surface Area ==
Line 1396: Line 1388:

Note that this works in Sage cell, but causes a zip file error in Jupyter
Line 1463: Line 1457:
== Parametric surface example (FIXME in Jupyter) == == Parametric surface example ==
Line 1465: Line 1459:

Note that this works in Sage cell, but causes a zip file error in Jupyter.
Line 1513: Line 1509:
    u(t) = u
    v(t) = v
    w(t) = w

Sage Interactions - Calculus

goto interact main page

Contents

  1. Sage Interactions - Calculus
    1. Root Finding Using Bisection
    2. Newton's Method
    3. A contour map and 3d plot of two inverse distance functions
    4. A simple tangent line grapher
    5. Numerical integrals with the midpoint rule
    6. Numerical integrals with various rules
    7. Some polar parametric curves
    8. Function tool
    9. Newton-Raphson Root Finding
    10. Coordinate Transformations
    11. Taylor Series
    12. Illustration of the precise definition of a limit
    13. A graphical illustration of sin(x)/x -> 1 as x-> 0
    14. Quadric Surface Plotter
    15. The midpoint rule for numerically integrating a function of two variables
    16. Gaussian (Legendre) quadrature
    17. Vector Calculus, 2-D Motion
    18. Vector Calculus, 3-D Motion
    19. Multivariate Limits by Definition
    20. Directional Derivatives
    21. 3D graph with points and curves
    22. Approximating function in two variables by differential
    23. Taylor approximations in two variables
    24. Volumes over non-rectangular domains
    25. Lateral Surface Area
    26. Parametric surface example
    27. Line Integrals in 3D Vector Field

Root Finding Using Bisection

by William Stein

bisect.png

Newton's Method

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

by William Stein

https://cloud.sagemath.com/projects/19575ea0-317e-402b-be57-368d04c113db/files/pub/2801-2901/2824-Double%20Precision%20Root%20Finding%20Using%20Newton's%20Method.sagews

newton.png

A contour map and 3d plot of two inverse distance functions

by William Stein

https://cloud.sagemath.com/projects/19575ea0-317e-402b-be57-368d04c113db/files/pub/2801-2901/2823.sagews

mountains.png

A simple tangent line grapher

by Marshall Hampton

tangents.png

Numerical integrals with the midpoint rule

by Marshall Hampton

num_int.png

Numerical integrals with various rules

by Nick Alexander (based on the work of Marshall Hampton)

num_int2.png

Some polar parametric curves

by Marshall Hampton. This is not very general, but could be modified to show other families of polar curves.

polarcurves1.png

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.

funtool.png

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.

newtraph.png

Coordinate Transformations

by Jason Grout

coordinate-transform-1.png coordinate-transform-2.png

Taylor Series

by Harald Schilly

taylor_series_animated.gif

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.

snapshot_epsilon_delta.png

A graphical illustration of sin(x)/x -> 1 as x-> 0

by Wai Yan Pong

sinelimit.png

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.

quadrics.png

The midpoint rule for numerically integrating a function of two variables

by Marshall Hampton

numint2d.png

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.

quadrature1.png quadrature2.png

Vector Calculus, 2-D Motion

By Rob Beezer

A fast_float() version is available in a worksheet

motion2d.png

Vector Calculus, 3-D Motion

by Rob Beezer

Available as a worksheet

motion3d.png

Multivariate Limits by Definition

by John Travis

http://sagenb.mc.edu/home/pub/97/

3D_Limit_Defn.png

3D_Limit_Defn_Contours.png

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

3Dgraph_with_points.png

Approximating function in two variables by differential

by Robert Marik

3D_differential.png

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).

taylor-3d.png

Volumes over non-rectangular domains

by John Travis

https://cloud.sagemath.com/projects/19575ea0-317e-402b-be57-368d04c113db/files/pub/2801-2901/2829.sagews

3D_Irregular_Volume.png

Lateral Surface Area

by John Travis

http://sagenb.mc.edu/home/pub/89/

Note that this works in Sage cell, but causes a zip file error in Jupyter

Lateral_Surface.png

Parametric surface example

by Marshall Hampton

Note that this works in Sage cell, but causes a zip file error in Jupyter.

parametric_surface.png

Line Integrals in 3D Vector Field

by John Travis

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

3D_Line_Integral.png

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