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# Blog - fluid flows and infinite dimensional manifolds (part 2)

Created stub for

Unfortunately I have to run to an important meeting and cannot add any content right now!

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We want content!

Comment Source:We want _content!_
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The problem is that several other people also demand content, and they take precedence because they pay for my wage :-)

Comment Source:<img src="http://math.ucr.edu/home/baez/emoticons/swimming.gif" alt="Swimming" /> The problem is that several other people also demand content, and they take precedence because they pay for my wage :-)
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Yup.

I have an idea of writing a series of posts starting with the Navier-Stokes equations, then simplifying them in various ways, getting down to the models that can be studied using infinite-dimensional groups. This could meet you half-way...

Comment Source:Yup. I have an idea of writing a series of posts starting with the Navier-Stokes equations, then simplifying them in various ways, getting down to the models that can be studied using infinite-dimensional groups. This could meet you half-way...
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edited March 2012

Sure, sounds like a good idea. I don't have a specific idea of what the next post should be about, right now. Maybe it should start with the promised derivation of the Kdv-Equation.

Comment Source:Sure, sounds like a good idea. I don't have a specific idea of what the next post should be about, right now. Maybe it should start with the promised derivation of the Kdv-Equation.
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edited April 2012

I added two paragraphs about the definition of incompressible flows and volume preserving diffeomorphisms and ideal fluids. We'll need both anyway, no matter how the rest of the posts will proceed, I think.

BTW, does anyone have access to this paper: Ebin, David G.; Marsden: Jerrold Groups of diffeomorphisms and the motion of an incompressible fluid. ?

I would like to better understand the role Sobolev spaces play in this whole topic, and everybody and his cat seems to refer to this one paper.

Edit: Got it, thanks Frederik!

Comment Source:I added two paragraphs about the definition of incompressible flows and volume preserving diffeomorphisms and ideal fluids. We'll need both anyway, no matter how the rest of the posts will proceed, I think. BTW, does anyone have access to this paper: Ebin, David G.; Marsden: <a href="http://www.ams.org/mathscinet-getitem?mr=271984">Jerrold Groups of diffeomorphisms and the motion of an incompressible fluid. </a>? I would like to better understand the role Sobolev spaces play in this whole topic, and everybody and his cat seems to refer to this one paper. Edit: Got it, thanks Frederik!
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I was going to ask what a Jerrold group is... but that must be a typo! Jerrold is Marsden's first name. By the way, he died fairly recently.

Comment Source:I was going to ask what a Jerrold group is... but that must be a typo! Jerrold is Marsden's first name. By the way, he died fairly recently.
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edited April 2012

Uh, yes, that's Mardsen, Jerrold: Groups of...

Sad to hear that he has passed away, maybe we could do a blog post in memoriam. When I learned about differential forms in my second year at the university of Göttingen, I asked the professor for a book about applications of this seemingly esoteric topic, and he named Marsden's "Foundations of Mechanics". So that was my first encounter with abstract differential geometry in theoretical physics...

Comment Source:Uh, yes, that's Mardsen, Jerrold: Groups of... Sad to hear that he has passed away, maybe we could do a blog post in memoriam. When I learned about differential forms in my second year at the university of Göttingen, I asked the professor for a book about applications of this seemingly esoteric topic, and he named Marsden's "Foundations of Mechanics". So that was my first encounter with abstract differential geometry in theoretical physics...
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Foundations of Mechanics is a great book. The four kings of symplectic geometry in the US were Guillemin and Sternberg on the east coast and Marsden and Weinstein on the west. Marsden is the one I knew least well - basically not at all!

Comment Source:_Foundations of Mechanics_ is a great book. The four kings of symplectic geometry in the US were Guillemin and Sternberg on the east coast and Marsden and Weinstein on the west. Marsden is the one I knew least well - basically not at all!
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9.

I think I have a storyline for this blog post:

• explain "ideal" and "incompressible",
• explain Euler's equations for ideal, incompressible and homogeneous fluids,
• show how to define a Hamiltonian structure that reproduces Euler's equation.

The stuff is all there now, it just needs to be better explained :-)

Comment Source:I think I have a storyline for this blog post: - explain "ideal" and "incompressible", - explain Euler's equations for ideal, incompressible and homogeneous fluids, - show how to define a Hamiltonian structure that reproduces Euler's equation. The stuff is all there now, it just needs to be better explained :-)
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• explain "ideal" and "incompressible",

So you want to study dry water according to von Neumann ;-)

Comment Source:> - explain "ideal" and "incompressible", So you want to study dry water according to von Neumann ;-)
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edited May 2012

Every model is an approximation, man...our own hallucination of what we use to call "reality" in everyday language is no exception.

Besides I do mention that an "ideal" fluid cannot form eddies and therefore is very ideal in the sense of an oversimplification.

Comment Source:Every model is an approximation, man...our own hallucination of what we use to call "reality" in everyday language is no exception. Besides I do mention that an "ideal" fluid cannot form eddies and therefore is very ideal in the sense of an oversimplification.
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edited April 2012

Yes, sure, man, I didn't except you to take it up as a serious critique (hence the ";-)")

I simply liked this quote of Feynman in his lectures - and he devotes a whole chapter to dry water - so I thought it could make a nice joke [in your blog post].

But perhaps I shouldn't read your reply as being too critical for my (constructively meant) comment, just because of my personal impression that you took my comment as being too critical, since that impression of mine might be a hallucination too.

Comment Source:Yes, sure, man, I didn't except you to take it up as a serious critique (hence the ";-)") I simply liked this quote of Feynman in his lectures - and he devotes a whole chapter to dry water - so I thought it could make a nice joke [in your blog post]. But perhaps I shouldn't read your reply as being too critical for my (constructively meant) comment, just because of my personal impression that you took my comment as being too critical, since that impression of mine might be a hallucination too.
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edited April 2012

...I didn't except you to take it up as a serious critique...

Why not? I think it is a very important topic to think and tell your audience about what kind of simplifications you make. In this case it is very important for me to explain that "ideal" excludes shear forces and therefore the possibility of forming eddies without external forces.

...so I thought it could make a nice joke...

Maybe, but I have to admit that I don't understand that quote. Incompressible means we cannot model shock waves, ideal means we cannot model eddy creation or decay, but besides that stratified flows can be modelled quite well. And although I don't know much about this topic, I think that historically aviation engineers have profited a lot from calculations performed with these simplifications.

So, in what sense, is an ideal incompressible fluid "dry"? (I know one should not explain jokes, but...)

Comment Source:<blockquote> <p> ...I didn't except you to take it up as a serious critique... </p> </blockquote> Why not? I think it is a very important topic to think and tell your audience about what kind of simplifications you make. In this case it is very important for me to explain that "ideal" excludes shear forces and therefore the possibility of forming eddies without external forces. <blockquote> <p> ...so I thought it could make a nice joke... </p> </blockquote> Maybe, but I have to admit that I don't understand that quote. Incompressible means we cannot model shock waves, ideal means we cannot model eddy creation or decay, but besides that stratified flows can be modelled quite well. And although I don't know much about this topic, I think that historically aviation engineers have profited a lot from calculations performed with these simplifications. So, in what sense, is an ideal incompressible fluid "dry"? (I know one should not explain jokes, but...)
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The way I understood the joke it is that "dry water" doesn't feel wet like water should do, it doesn't stick, so it doesn't feel real.

But, as it appears, since then dry water has been invented and has a different meaning now, just cancel the joke...

have profited a lot from calculations performed with these simplifications

yes, apart from regions where there's a boundary layer or a turbulent wake, potential flow is a fine approximation, as far as I know. (Btw, something I've been wondering about, do you know if there are cases where the vorticity is nonzero but the flow can be assumed ideal and incompressible nevertheless) In this respect, the different boundary conditions for tangential velocity between ideal and non-ideal fluids are intruiging too.

Comment Source:The way I understood the joke it is that "dry water" doesn't feel wet like water should do, it doesn't stick, so it doesn't feel real. But, as it appears, since then [dry water](http://en.wikipedia.org/wiki/Dry_water) has been invented and has a different meaning now, just cancel the joke... > have profited a lot from calculations performed with these simplifications yes, apart from regions where there's a boundary layer or a turbulent wake, potential flow is a fine approximation, as far as I know. (Btw, something I've been wondering about, do you know if there are cases where the vorticity is nonzero but the flow can be assumed ideal and incompressible nevertheless) In this respect, the different boundary conditions for tangential velocity between ideal and non-ideal fluids are intruiging too.
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edited April 2012

The way I understood the joke it is that "dry water" doesn't feel wet like water should do, it doesn't stick, so it doesn't feel real.

Oh, well, it would still cool you when it evaporates and it will also carry you when you swim in it. And it will also run down in droplets on you with finite velocity, even if you don't feel it.

But, as it appears, since then dry water has been invented and has a different meaning now, just cancel the joke...

Btw, something I've been wondering about, do you know if there are cases where the vorticity is nonzero but the flow can be assumed ideal and incompressible nevertheless.

I'm not sure I understand your question: You can start with any vorticity that you like, as an initial condition. Do you ask for any interesting real situations that can be modelled this way? Then I don't know of any...

Comment Source:<blockquote> <p> The way I understood the joke it is that "dry water" doesn't feel wet like water should do, it doesn't stick, so it doesn't feel real. </p> </blockquote> Oh, well, it would still cool you when it evaporates and it will also carry you when you swim in it. And it will also run down in droplets on you with finite velocity, even if you don't feel it. <blockquote> <p> But, as it appears, since then dry water has been invented and has a different meaning now, just cancel the joke... </p> </blockquote> <img src="http://math.ucr.edu/home/baez/emoticons/boing.gif" alt="boing" /> <blockquote> <p> Btw, something I've been wondering about, do you know if there are cases where the vorticity is nonzero but the flow can be assumed ideal and incompressible nevertheless. </p> </blockquote> I'm not sure I understand your question: You can start with any vorticity that you like, as an initial condition. Do you ask for any interesting real situations that can be modelled this way? Then I don't know of any...
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edited April 2012

My (amateur) understanding is that Von Neumann, and most physicists at the time, saw turbulence in flowing water as almost the defining thing to be interested in about flowing water, which as you've said isn't within the scope of the simplified system model that you're using. So "dry water" is in the sense of "the water which remains after you remove the very thing that makes it watery".

(This isn't ctricising your usage of the model, just explaining what I understand the joke was.)

Comment Source:My (amateur) understanding is that Von Neumann, and most physicists at the time, saw turbulence in flowing water as almost the defining thing to be interested in about flowing water, which as you've said isn't within the scope of the simplified system model that you're using. So "dry water" is in the sense of "the water which remains after you remove the very thing that makes it watery". (This isn't ctricising your usage of the model, just explaining what I understand the joke was.)
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Tim wrote:

You can start with any vorticity that you like, as an initial condition. Do you ask for any interesting real situations that can be modelled this way? Then I don't know of any...

Thanks!

it will also run down in droplets

I'm not sure about that, since the tangential velocity could be anything, there's no friction between the fluid and the surface.

Comment Source:Tim wrote: > You can start with any vorticity that you like, as an initial condition. Do you ask for any interesting real situations that can be modelled this way? Then I don't know of any... Thanks! > it will also run down in droplets I'm not sure about that, since the tangential velocity could be anything, there's no friction between the fluid and the surface.
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edited April 2012

I renamed the page to Blog - fluid flows and infinite dimensional manifolds II but Instiki managed to keep the old page as well, somehow. Well, nevermind, I changed the storyline a little bit, which is now simply:

1) explain "ideal",

2) explain "incompressible",

3) explain how Euler's equation comes up as the geodesic equation of the Lie group of volume preserving diffeomorphisms in $\mathbb{R}^3$.

Seems to be enough stuff for one blog post. I think I need to fix a few points at the end of the post, but besides that it is pretty much done.

Comment Source:I renamed the page to [[Blog - fluid flows and infinite dimensional manifolds II]] but Instiki managed to keep the old page as well, somehow. Well, nevermind, I changed the storyline a little bit, which is now simply: 1) explain "ideal", 2) explain "incompressible", 3) explain how Euler's equation comes up as the geodesic equation of the Lie group of volume preserving diffeomorphisms in $\mathbb{R}^3$. Seems to be enough stuff for one blog post. I think I need to fix a few points at the end of the post, but besides that it is pretty much done.
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Ok, I'm done for now, I think the post is done so far, feedback is welcome. Especially one question: Is it too short or too long?

@John: I remember that you had a list of points every author should keep in mind (formatting and stuff), but I did not find it. Maybe we should put it on the page with the list of blog posts on the Wiki.

Comment Source:Ok, I'm done for now, I think the post is done so far, feedback is welcome. Especially one question: Is it too short or too long? @John: I remember that you had a list of points every author should keep in mind (formatting and stuff), but I did not find it. Maybe we should put it on the page with the list of blog posts on the Wiki.
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GREAT! I'll check out your post, Tim. By the way, everyone should read Tim's post and take it as an example of the kind of thing I want more of on the blog. Join the fun!

As for formatting, I forget where I put that advice, so my quick advice is to take some of your blog posts on the Wiki that I already edited, like A quantum of warmth and Eddy who?, and make all the formatting like that!

But maybe when I read your new post I'll generate a new list of grumpy formatting comments, and try to put somewhere useful this time.

Comment Source:GREAT! I'll check out your post, Tim. By the way, everyone should read Tim's post and take it as an example of the kind of thing I want more of on the blog. Join the fun! As for formatting, I forget where I put that advice, so my quick advice is to take some of your blog posts on the Wiki that I already edited, like [A quantum of warmth](http://www.azimuthproject.org/azimuth/show/Blog+-+a+quantum+of+warmth) and [Eddy who?](http://www.azimuthproject.org/azimuth/show/Blog+-+eddy+who%3F), and _make all the formatting like that!_ But maybe when I read your new post I'll generate a new list of grumpy formatting comments, and try to put somewhere useful this time.
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Tim wrote:

I renamed the page to Blog - fluid flows and infinite dimensional manifolds II but Instiki managed to keep the old page as well, somehow.

I believe that's the famous 'cache bug' which Andrews talks about. The old one should go away when the Midgard serpent wakes up, wants a snack, and eats it.

Comment Source:Tim wrote: > I renamed the page to [Blog - fluid flows and infinite dimensional manifolds II](http://www.azimuthproject.org/azimuth/show/Blog%20-%20fluid%20flows%20and%20infinite%20dimensional%20manifolds%20II) but Instiki managed to keep the old page as well, somehow. I believe that's the famous 'cache bug' which Andrews talks about. The old one should go away when the Midgard serpent wakes up, wants a snack, and eats it. <img src = "http://images4.fanpop.com/image/photos/19400000/Midgard-serpent-norse-mythology-19478009-508-614.jpg" alt = ""/>
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edited May 2012

Okay, here are some comments, written as I edit. Nobody should feel the need to remember all these things, or avoid writing blog articles because of them! I just feel like listing formatting rules that I've adopted on Azimuth, and the easiest way to remember them is to see Tim breaking them.

1) I'll break up some long sentences, since English speakers, and especially people reading blogs instead of books, tend to fall asleep around the third clause.

2) On Azimuth I religiously write things like this:

but---as could be expected---assuming a little bit more of a mathematical background

not this:

but - as could be expected - assuming a little bit more of a mathematical background

That's because publishers like the former (it comes out looking nice on Azimuth, not here). I'm going to publish the 'network theory' notes... and, who knows, we may publish more of this stuff too someday. But even if we don't consistency is reassuring to the reader.

3) Section headers should be like this:

<h3>What is an ideal fluid?</h3>

not this:

<h4>What is an ideal fluid?</h4>

The latter come out almost indistinguishable from normal-font boldface text.

4) Defined terms should be boldface (not quoted, italicized, etc.). E.g.:

external forces like gravity that are there whether or not your fellow fluid parcels surround you or are absent,

This is not quite a mathematically precise definition, but that's okay: the boldface lets the reader know that this is a word being explained, not one they're supposed to know.

Comment Source:Okay, here are some comments, written as I edit. Nobody should feel the need to remember all these things, or avoid writing blog articles because of them! I just feel like listing formatting rules that I've adopted on Azimuth, and the easiest way to remember them is to see Tim breaking them. <img src = "http://math.ucr.edu/home/baez/emoticons/tongue.gif" alt = ""/> 1) I'll break up some long sentences, since English speakers, and especially people reading blogs instead of books, tend to fall asleep around the third clause. 2) On Azimuth I religiously write things like this: but---as could be expected---assuming a little bit more of a mathematical background not this: but - as could be expected - assuming a little bit more of a mathematical background That's because publishers like the former (it comes out looking nice on Azimuth, not here). I'm going to publish the 'network theory' notes... and, who knows, we may publish more of this stuff too someday. But even if we don't consistency is reassuring to the reader. 3) Section headers should be like this: <h3>What is an ideal fluid?</h3> not this: <h4>What is an ideal fluid?</h4> The latter come out almost indistinguishable from normal-font boldface text. 4) Defined terms should be boldface (not quoted, italicized, etc.). E.g.: > <b>external</b> forces like gravity that are there whether or not your fellow fluid parcels surround you or are absent, This is not quite a mathematically precise definition, but that's okay: the boldface lets the reader know that this is a word being explained, not one they're supposed to know.
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edited May 2012

5) Pictures should be formatted like this:

<div align = "center"><img width = "450" src = "http://www.azimuthproject.org/azimuth/files/pressure_in_ideal_fluid.png" alt = "pressure in an ideal fluid"/></div>

and with a link if there's a source for the picture - especially for graphs of climate information and the like. Tim did this one exactly right!

6) References: for books, don't use

&bull; Alexandre Chorin, Jerrold E. Marsden, "A Mathematical Introduction to Fluid Mechanics", 3rd edition, Springer, New York 1993.

&bull; Alexandre Chorin and Jerrold E. Marsden, _A Mathematical Introduction to Fluid Mechanics_, 3rd edition, Springer, New York 1993.

Our conventions on references are on the [How to] page on the Azimuth Wiki.

7) Tim writes:

If you are a parcel of an incompressible fluid, this means that your volume does not change over time.'

I really like this use of "you", even though it's a bit extreme. Using "you", while informal, gets people involved. I now feel like a parcel of incompressible fluid. How will I ever lose weight?

8) Tim wrote:

Assuming that we have a fixed <b>volume form</b> $\mu$

Since this term is not being defined right here, it should not be boldface. (Later it is defined.)

Comment Source:5) Pictures should be formatted like this: <div align = "center"><img width = "450" src = "http://www.azimuthproject.org/azimuth/files/pressure_in_ideal_fluid.png" alt = "pressure in an ideal fluid"/></div> and with a link if there's a source for the picture - especially for graphs of climate information and the like. Tim did this one exactly right! 6) References: for books, don't use &bull; Alexandre Chorin, Jerrold E. Marsden, "A Mathematical Introduction to Fluid Mechanics", 3rd edition, Springer, New York 1993. but instead &bull; Alexandre Chorin and Jerrold E. Marsden, _A Mathematical Introduction to Fluid Mechanics_, 3rd edition, Springer, New York 1993. Our conventions on references are on the [[How to]](http://www.azimuthproject.org/azimuth/show/How+to#References) page on the Azimuth Wiki. 7) Tim writes: > If you are a parcel of an incompressible fluid, this means that your volume does not change over time.' I really like this use of "you", even though it's a bit extreme. Using "you", while informal, gets people involved. I now feel like a parcel of incompressible fluid. How will I ever lose weight? 8) Tim wrote: Assuming that we have a fixed <b>volume form</b> $\mu$ Since this term is not being defined right here, it should not be boldface. (Later it _is_ defined.)
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Thanks. This weekend I will compile all of this somewhere on the Wiki, or else...

Comment Source:Thanks. This weekend I will compile all of this somewhere on the Wiki, or else...
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Great. I'm not done yet - I got about halfway through editing your blog article; I'll try to finish up now. I've made lots of changes in your writing not mentioned above, so please take a look and okay what I've done before I post this article. It looks great!

Comment Source:Great. I'm not done yet - I got about halfway through editing your blog article; I'll try to finish up now. I've made lots of changes in your writing not mentioned above, so please take a look and okay what I've done before I post this article. It looks great!
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edited May 2012

9) It will save me time if math text that needs to be roman is written as, say

$\mathrm{div}(X)$

rather than

$div(X)$

since while the wiki and this forum have the peculiar feature that multiletter symbols in math mode are automatically converted to roman font, this is not true of most TeX installations, e.g. my blog or MathJaX. Of course for lots of standard symbols like \sin, \cos etcetera a backslash is enough. Maybe it even works for \div. But div' by itself won't work.

10) Much more importantly, it will save me time if displayed equations are written as

$$E = mc^2$$

without linebreaks, rather than say

$$ E = mc^2 $$

as Tim is now doing it (roughly). The reason is that I need to convert them to

$latex E = mc^2$

to work on the blog - and not being very clever at these things, I find it easier to do this semi-automatically if we start with a format where the initial SS is clearly different than the final one!

Comment Source:More comments: 9) It will save me time if math text that needs to be roman is written as, say $\mathrm{div}(X)$ rather than $div(X)$ since while the wiki and this forum have the peculiar feature that multiletter symbols in math mode are automatically converted to roman font, this is not true of most TeX installations, e.g. my blog or MathJaX. Of course for lots of standard symbols like \sin, \cos etcetera a backslash is enough. Maybe it even works for \div. But div' by itself won't work. 10) **Much more importantly**, it will save me time if displayed equations are written as $$E = mc^2$$ without linebreaks, rather than say $$ E = mc^2 $$ as Tim is now doing it (roughly). The reason is that I need to convert them to $latex E = mc^2$ to work on the blog - and not being very clever at these things, I find it easier to do this semi-automatically if we start with a format where the initial SS is clearly different than the final one!
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edited May 2012

11) If you have a long line of equations, it's better to use \begin{array} to line them up vertically instead of letting them stretch across the page horizontally. One reason is that the blog is only 450 pixels wide, and equations that are too long just disappear at the right margin! Another is that it's easier to follow the steps of reasoning when you can see similar stuff above similar stuff.

You can see a lot of examples in Blog - fluid flows and infinite-dimensional manifolds, and peek into that page to see how they're done.

Comment Source:11) If you have a long line of equations, it's better to use \begin{array} to line them up vertically instead of letting them stretch across the page horizontally. One reason is that the blog is only 450 pixels wide, and equations that are too long just disappear at the right margin! Another is that it's easier to follow the steps of reasoning when you can see similar stuff above similar stuff. You can see a lot of examples in [[Blog - fluid flows and infinite-dimensional manifolds]], and peek into that page to see how they're done.
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28.

Okay, here are two comments on the substance of what you wrote, Tim:

1) I think you should have reminded the reader of Euler's equation at the end and make it clear that we've really gotten Euler's equation. It's worth a little celebration! So, I changed this part a little.

2) I'm worried about these steps:

$$- \int_M \langle v_x, \nabla \times (u_x \times w_x) \rangle \;d \mu(x) = - \int_M \langle \nabla \times v_x, u_x \times w_x \rangle \;d \mu(x) = - \int_M \langle (\nabla \times v_x) \times u_x, w_x \rangle \;d \mu(x)$$ Could there be a sign mistake in each step? Usually moving a derivative from one side to the other introduces a minus sign, thanks to integration by parts. But maybe there's another minus sign in the next step.

Comment Source:Okay, here are two comments on the _substance_ of what you wrote, Tim: 1) I think you should have reminded the reader of Euler's equation at the end and make it clear that we've really _gotten_ Euler's equation. It's worth a little celebration! So, I changed this part a little. 2) I'm worried about these steps: $$- \int_M \langle v_x, \nabla \times (u_x \times w_x) \rangle \;d \mu(x) = - \int_M \langle \nabla \times v_x, u_x \times w_x \rangle \;d \mu(x) = - \int_M \langle (\nabla \times v_x) \times u_x, w_x \rangle \;d \mu(x)$$ Could there be a sign mistake in each step? Usually moving a derivative from one side to the other introduces a minus sign, thanks to integration by parts. But maybe there's another minus sign in the next step.
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Okay, Tim. Look at the article, see if you approve of the changes I've made, and let me know!

Comment Source:Okay, Tim. Look at the article, see if you approve of the changes I've made, and let me know!
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I think the 'dry water' joke would be good to include and explain, since it's about how Euler's equation is oversimplified. I don't think you should take the word 'dry' very literally: 'dry water' simply means 'water that's not like real water'.

By the way: I've been using "part 1", "part 2" etc. for multi-part series on the Azimuth Blog, not roman numerals I, II. So, I changed the title to:

where of course the word 'Blog -' won't show up in the final article.

Comment Source:I think the subtitle of this article is less witty than usual, Tim. How about 'Dry water' or "In an ideal universe' or something? I think the 'dry water' joke would be good to include and explain, since it's about how Euler's equation is oversimplified. I don't think you should take the word 'dry' very literally: 'dry water' simply means 'water that's not like real water'. By the way: I've been using "part 1", "part 2" etc. for multi-part series on the Azimuth Blog, not roman numerals I, II. So, I changed the title to: * [[Blog - fluid flows and infinite dimensional manifolds (part 2)]] where of course the word 'Blog -' won't show up in the final article.
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31.

John wrote:

2) I'm worried about these steps:...

I've expanded that part so that it should be clear now that and why there is not a sign error - unless I made a mistake, of course :-) But since I explained divergence and all the rest of the vector analysis stuff, I should have been more explicit there, of course, from the beginning.

Sigh. I have included the "dry water joke", but mostly to explain why I don't think it is much to the point :-) With all due respect to John von Neumann :-)

Well, I think we can call it done now.

Comment Source:John wrote: <blockquote> <p> 2) I'm worried about these steps:... </p> </blockquote> I've expanded that part so that it should be clear now that and why there is not a sign error - unless I made a mistake, of course :-) But since I explained divergence and all the rest of the vector analysis stuff, I should have been more explicit there, of course, from the beginning. <blockquote> <p> I think the subtitle of this article is less witty than usual, Tim. How about 'Dry water' or "In an ideal universe' or something? </p> </blockquote> Sigh. I have included the "dry water joke", but mostly to explain why I don't think it is much to the point :-) With all due respect to John von Neumann :-) Well, I think we can call it done now.
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edited May 2012

Okay, I'll post it now. It'll take a while to transfer it into blog-ready format. By the way, you forgot this comment of mine from before:

Much more importantly, it will save me time if displayed equations are written as

$$E = mc^2$$

without linebreaks, rather than say

$$ E = mc^2 $$

as Tim is now doing it (roughly). The reason is that I need to convert them to

$latex E = mc^2$

Comment Source:Okay, I'll post it now. It'll take a while to transfer it into blog-ready format. By the way, you forgot this comment of mine from before: **Much more importantly**, it will save me time if displayed equations are written as $$E = mc^2$$ without linebreaks, rather than say $$ E = mc^2 $$ as Tim is now doing it (roughly). The reason is that I need to convert them to $latex E = mc^2$
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33.

Okay, the post is here now! Thanks a million, Tim. I learned a lot from this one.

Comment Source:Okay, the post is [here](http://johncarlosbaez.wordpress.com/2012/05/12/fluid-flows-and-infinite-dimensional-manifolds-part-2/) now! Thanks a million, Tim. I learned a lot from this one.
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34.

Much more importantly, it will save me time if displayed equations are written as...

Sorry, that's why I need all these points in form of a checklist, so that I can go through it one by one. I'll try to collect them somewhere...

Comment Source:<blockquote> <p> Much more importantly, it will save me time if displayed equations are written as... </p> </blockquote> Sorry, that's why I need all these points in form of a checklist, so that I can go through it one by one. I'll try to collect them somewhere...