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FLOWS IN FLATLAND: A ROMANCE OF FEW DIMENSIONS PART I

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In this paper, we present our general results about non-vanishing gradient-like vector flows on $n$-dimensional manifolds with boundary in the context of the flows on $2$-dimensional surfaces with boundary. We take advantage of the relative simplicity of $2$-dimensional worlds to explain and popularize our approach to the Morse theory on smooth manifolds with boundary. In this approach, the boundary effects take the central stage.

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Please take a look at the important information in this header. We encourage you to keep this file on your own disk, keeping an electronic path open for the next readers. Do not remove this. Corrected EDITIONS of our etexts get a new NUMBER, flat11.txt VERSIONS based on separate sources get new LETTER, flat10a.txt We are now trying to release all our books one month in advance of the official release dates, for time for better editing. The official release date of all Project Gutenberg Etexts is at Midnight, Central Time, of the last day of the stated month. A preliminary version may often be posted for suggestion, comment and editing by those who wish to do so. To be sure you have an up to date first edition [xxxxx10x.xxx] please check file sizes in the first week of the next month. Since our ftp program has a bug in it that scrambles the date [tried to fix and failed] a look at the file size will have to do, but we will try to see a new copy has at least one byte more or less. Information about Project Gutenberg (one page) We produce about two million dollars for each hour we work. The fifty hours is one conservative estimate for how long it we take to get any etext selected, entered, proofread, edited, copyright searched and analyzed, the copyright letters written, etc. This projected audience is one hundred million readers. If our value per text is nominally estimated at one dollar, then we produce 2 million dollars per hour this year we, will have to do four text files per month: thus upping our productivity from one million.
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