From Math Images
Yeah, as I look at it, I think your approach is really good. This page is just about done, but if you go through the checklist Anna and I sent you, you'll see you didn't address one of the items. See if you can figure out which one it is! How fun!
Oh, and can you change "infinitely subregion" to "infinitely small subregion" or "infinitesimal subregion"?
I edited my explanations a bit. I still left the infinitesimal- box explanation after the equation; I think it makes sense to introduce the idea then explain it with the box diagram. I did notice that I said 'summing divergence over the entire volume' a few times, so I trimmed that down some.
Nice edits. Everything you say is really good. I would suggest a couple of organizational changes.
- The divergence of a vector field is a measurement of the expansion or contraction of the field; if more water is being introduced then the divergence is positive.
This isn't really an organizational change, but this definition doesn't make it clear that divergence has a different value at each point in space. How about:
- The divergence of a vector field at a point measures the net flow of water going outwards from that point.
- Summing up divergence over the entire volume means we sum the flow into or out of each infinitely subregion, since a flow into one infinitesimal subregion means flow out of an adjacent subregion, which effects the next adjacent subregion, and so on until the boundary of the entire volume is reached.
Beautifully written. The only problem is that you use the phrase "summing divergence over the entire volume" several sentences before you actually write this description. What you could do is write the above paragraph before you state the equation or first use that phrase after you state the equation. Either way, it should stay in the page somehow.
Also, the "since" is a little bit misleading, because the truth of the first part of the sentence is totally independent of the stuff that follows "since." You could write:
- Summing up divergence over the entire volume means we sum the flow into or out of each infinitely subregion. A flow into one infinitesimal subregion means flow out of an adjacent subregion, which effects the next adjacent subregion, and so on until the boundary of the entire volume is reached, and therefore [something about how the inward and outward flows cancel each other out, so the sum of all these flows equals the total flux through the boundary].
First, one small detail:
- Flowing water can be considered a vector field because at each point the water has a position and a velocity vector.
Saying "point" and "position" is redundant. How about:
- Flowing water can be modeled with a vector field because at each point in space the water has a velocity vector.
Second, I like a lot of what you say in the more mathematical section, and the example you chose is really good. The big thing that's missing is actually connecting the idea of flux to the idea of divergence or to the divergence theorem, such as:
- An interpretation of divergence at a point as representing the flux per unit volume through a very small cube surrounding that point
- An interpretation of the left-hand side of the divergence theorem (for example, it's what you get if you divide the solid into lots of tiny cubes and add the total flux through each of those cubes)
- An interpretation of the right-hand side of the divergence theorem
- If you want to go crazy, a plausibility explanation for why these two quantities should be equal (for example, every source of outward/positive flux through one tiny cube is a source of inward/negative flux for an adjacent tiny cube; the only flux effects that aren't canceled out this way are the ones going through the boundary of the solid, because there is no adjacent cube)
- Also, images to go with all these interpretations would be great. I know that's a lot to do and maybe you don't have time, but you're good at putting together coherent, image-rich content pretty quickly.
Third, I figured out how to say what I sort of got at in today's meeting. The following line is great:
- the total the fluid being introduced into a volume is equal to the total fluid flowing out of the boundary of the volume if the quantity of fluid in the volume is constant
You're right that it's very clear. The problem is that if you tell people something that's too obvious, it won't be interesting. Remember, if the person reading this page doesn't know multi-variable calculus, this first paragraph is the only one they will read, and I'm not sure it actually teaches them anything. I'm also not sure that it fails, so maybe it's fine and you can ignore this point completely.
The more detailed description is all one huge paragraph! is there any way you could break it up?