OK, so, yes, I agree that the current flow of the page is very nice. However, I find it very hard to believe that it would be impossible to make the change that I am suggesting and still make the page flow well with the help of a bit of additional editing. Also, isn't making as much of the material as possible available to a wide range of readers a higher priority than not disturbing a nice page structure, especially given that the change I'm suggesting would have a reasonably significant impact on the depth of the page for non-calculus students?
I can understand that this may be more work than you want to deal with, and all of y'all's points are good. I just think you're under-prioritizing accessibility. But if you're determined to continue along this path, then sure, I guess Anna's suggested middle ground will work.
I actually feel like moving that information would require too much restructuring of the page, and I like the way if flows (no pun intended).
I might add a sentence after this one "The collection of vectors is known as a vector field." saying something about the fluid moving faster and in a different direction in the middle of the picture.
Does that sound like a middle ground?
I agree about the turbulence. I didn't actually mean to say that you should get rid of that part, sorry if it seemed like I was saying that.
The thing about your basic description is that it currently describes what the vector field represents, but it doesn't give the reader any way to interpret or analyze the vector field. By putting all the flow-line analysis in the more mathematical section, you don't allow any reader who is scared off by multi-variable calculus to learn about this way of interpreting vector fields.
I totally understand your concern about the basic description getting too long. What if you move this material to the basic description but have it hidden by default under a subheading called "Analysis of main image" or something.
I like your parametrized path addition. Can you explicitly state that x(t) in the differential equation is the parametrized path, and add a vector symbol thing on top of the x?
I removed the word overlapping, but the mention of turbulence was one of Steve Maurer's suggestions, so I'll leave it in. I added a short bit about the analytical definition of flow lines to make it a little 'more mathematical'. I think I'll leave the two sections separated as they are; I don't want the basic description to be too long.
As usual, good descriptions and excellent paired images. A couple of suggestions.
Almost everything in the more mathematical description can be moved to the basic description, in particular, everything about flow lines.
//The center portion of the fluid is more turbulent, as vectors point towards each other and overlap.//
I don't think the overlapping is significant to the turbulence, as this overlapping is a result of how densely you choose to show arrows as much as it is about anything else.
Might you want to redirect the user back to the applet when you talk about flow lines? I'd understand doing it or not doing it. Either way, the page is ready for the public in my opinion.
The colors, as far as I know, are purely aesthetic and do not relate to the mathematics of the vector field. As for the length of the vectors, I mention in the basic section that the "length of the arrow corresponds to how fast" something is moving, but I added another line clarifying that the direction of the vector is the direction the fluid is moving, and its length shows how fast the fluid is moving.
This is great stuff, I have one question though, do the different colors of the vectors in the main image mean anything? If they do, you should add that in somewhere. Also, you may want to add a line or two about the meaning of the lengths of the vectors, I am thinking specifically about the last image
"The instantaneous velocity field of a fluid." Is a gentler intro possible?
The vectors pictured represent the fluid's velocity at
certain points "the tails of the arrows shown" or something.
"Click for an applet that demonstrates behavior of objects in vector fields:" I'd give 'em a couple suggestions to try before dumping them into the nice applet.
More Mathematical: Take the first diagram and describe very briefly what the motion of the fluid is to confirm what most people will see.
You still clump too much stuff together. The next paragraph "The vector field ..." should not start till after the rotational vector field picture, and should go along with the flow line figure.
Your final paragraph should start after the flow line figue and maybe have some illustrative images?
Good stuff! We just have to get you to stretch things out more to help people read one concept at a time.
Check out the "flow" part of the java applet here http://math.la.asu.edu/~kawski/myjava/vfanalyzer/
You can click and "drop" a shape and see where it goes and how it stretches. It's pretty fun... and I was wondering if you wanted to link to it.
You can also embed it within the page if you want. Check out my lorenz attractor page.