Tag Archives: mesh

exposition, math, Mathematica, representations, understanding math

Improved clouds

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The interactive examples in this post require installing Wolfram CDF player, which is free and works on most desktop computers using Firefox, Safari and Internet Explorer, but not Chrome.

The source code for these demos is Animated Riemann.nb at my Mathematica Site. The notebook is is available for free use under a Creative Commons Attribution-ShareAlike 2.5 License. The notebook can be read by CDF Player if you cannot make the embedded versions in this post work.

The animated clouds show two hundred precalculated clouds for each picture, so you get the same clouds each time you run the animation.  It would have taken too long to generate the random clouds on the fly.  Each list of two hundred took about seven minutes to create on my computer.

In my post Riemann Clouds Improved I showed examples of clouds of values of Riemann sums in such a way that you could see the convergence to the value, the efficiency of the midpoint rule, and other things.  Here I include two Riemann sums that are shown

  • as manipulable graphs,
  • in clouds in an animated form.

Each manipulable graph (see Elaborate Riemann Sums Demo) has a slider to choose the mesh (1/n) of the partitions.  The small plus sign besides the slider gives you additional options. The buttons allow you to choose the type of partition and the type of evaluation points.

Each cloud shows a collection of values of random Riemann sums of the function, plotted by size of mesh (an upper bound on the width of the largest subdivision) and the value of the sum.  The cloud shows how the sums converge to the value of the integral. 

Every dot represents a random partition.  The sums with blue dots have random valuation points, the green dots use the left side of the subdivision, the brown dots the right side, and the red dots the midpoint.  The clouds may be suitable for students to study.  Some possible questions they could be asked to do are listed at the end.

Pressing the starter shows many clouds in rapid succession.  I don't know how much educational value that has but I think it is fun, and fun is worthwhile in itself.

Quarter Circle

Manipulable graph:

Animated cloud

 

Sine wave

Manipulable graph:

Animated cloud

Questions

I am not sure of the answers to some of these myself.

  • Why is the accuracy generally better for the sine wave than for the quarter circle?  
  • Why are the green dots above all the others and the brown dots below all the others in the quarter circle?
  • Why are they mixed in with the others for the sine curve?  In fact why do they tend upward? (Going from right to left, in other words in the direction of more accuracy).
  • Why are the midpoint sums so much more accurate?
  • Why do they tend downward for the sine wave?
  • Is it an optical illusion or do they also tend downward for the quarter circle? 

Notes:

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math, representations

Riemann clouds improved

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In my post Playing with Riemann Sums I showed a couple of clouds of points, each representing a particular Riemann sum for a particular function.   I have extended the code in a couple of ways.

The new code is in the Mathematica notebook and CDF file called MoreRiemann in the Mathematica section of abstractmath.   The .nb form is a Mathematica Notebook, which requires Mathematica to run and allows you to manipulate the objects and change the code in the notebook as you wish.  In particular, you can rerun the commands generating the clouds to get a new random result.  The .cdf file contains the same material and can be viewed using Mathematica CDF Player, which is available free here.  Both files have several other examples besides the ones shown below.

As always, my code is one-time code to show the ideas, but it is available freely via the Creative Commons Attribution – ShareAlike 3.0 License. I hope people will feel free to develop it further for use in teaching or for their own purposes.

Below is a cloud for \int_0^2 \sqrt{4-x^2} dx, the area of a quarter circle of radius 2, which is \pi.  The blue dots are arbitrary random Riemann sums with mesh shown on the horizontal axis and value on the vertical axis.  The partitions and the point in each subinterval are both random.  The red dots are arbitrary Riemann sums with random partitions but using the midpoint for value.

The next cloud shows random blue dots with the same meaning as above.  The red dots are Riemann sums with uniform subintervals evaluated at midpoints.  Possible discussion question for both of the clouds above:

  • Why do the red dots trend upward?

The following cloud is like the cloud above  with the addition of green dots representing uniform partitions evaluated at the left endpoint or right endpoint. (But the mesh scale is extended, giving different proportions to the picture.)

Of course the left endpoint gives the upper sums and the right endpoint gives the lower sums.

  • Explain the slight downward curvature of both green streaks.
  • Explain the big gap between the blue dots and the green dots.  (Requires some machinations with probability.)
  • Would there be blue dots a lot nearer the green dots if I ran the command asking for many more blue dots?

(These are idle questions I haven't thought about myself, but I'll bet they could be turned into good projects in analysis classes.)

Here is a cloud for \int_0^{\pi}\sin x dc with everything random for the blue dots and random partitions but midpoints for the red dots.

  • Why do these red dots trend upward?

The cloud below is for the same integral but uses uniform subintervals for the midpoint and adds green points for both the left endpoint and the right endpoint of uniform subinterval.

  • Why on earth do all the green dots trend downward???

This is a similar picture for \int_0^1 x^2 dx.  There are red dots but they are kind of drowned out.

And finally, here is \int_{\frac{1}{2}}^2 \frac{1}{x} dx:

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