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A Summer of Maths (ASoM) 2016

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    (Original post by liamm691)
    can anyone working through the vector and matrices help me with this example please using the equation of the line used in the attachments
    can i just rearrange and sub yw in?
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    (Original post by liamm691)
    can i just rearrange and sub yw in?
    If  z \bar{w} = \bar{z} w = \overline{z \bar{w}} then  z \bar{w} \in \mathbb{R}. Hence  z = k/\bar{w} for a real k. Can you finish it off?
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    (Original post by Gregorius)
    If  z \bar{w} = \bar{z} w = \overline{z \bar{w}} then  z \bar{w} \in \mathbb{Z}. Hence  z = k/\bar{w} for a real k. Can you finish it off?
    \mathbb{Z}? :curious:
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    (Original post by Zacken)
    \mathbb{Z}? :curious:
    Oh what an idiot

    Edited. I blame senility.
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    (Original post by Gregorius)
    Oh what an idiot

    Edited. I blame senility.
    Was half-hoping that there was some deep theorem that would make everything into integers. :lol:
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    (Original post by Gregorius)
    If  z \bar{w} = \bar{z} w = \overline{z \bar{w}} then  z \bar{w} \in \mathbb{R}. Hence  z = k/\bar{w} for a real k. Can you finish it off?
    ahhh yes thank you
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    (Original post by Zacken)
    Was half-hoping that there was some deep theorem that would make everything into integers. :lol:
    It would make life so much simpler. Even better if there were only finitely many of them too...
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    (Original post by Zacken)
    \mathbb{Z}? :curious:
    Well R,Yeh cause Im=0
    Y ?Is it wrong?


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    Anyone have any videos to recommend for learning Green's theorem/Stoke's theorem?
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    (Original post by Student403)
    Anyone have any videos to recommend for learning Green's theorem/Stoke's theorem?
    MIT YouTube channel might have something on it?


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    (Original post by Student403)
    Anyone have any videos to recommend for learning Green's theorem/Stoke's theorem?
    There are lectures on the MIT OpenCourseware for a large body of multivar and vector calculus concepts which are very good, and explain each concept in a lucid intuitive terms.
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    (Original post by Student403)
    Anyone have any videos to recommend for learning Green's theorem/Stoke's theorem?
    Take a look at section 3.6 of this chapter of the Feynman lectures, where he explains Stoke's theorem in a nice intuitive way:

    http://www.feynmanlectures.caltech.edu/II_03.html

    You would probably benefit from reading through chapters 2 and 3, where Feynman covers a lot of vector calculus from scratch.
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    (Original post by Student403)
    Anyone have any videos to recommend for learning Green's theorem/Stoke's theorem?
    Khanacademy explains the intuition of Green'/Stokes really well in my opinion. There is no rigour/proofs or anything but it served as an excellent introduction imo.
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    Any feedback on my argument for there being infinitely many rationals in the interval  [a,b] ? (a,b are any reals, b>a)
    Let  \epsilon = \dfrac{1}{b-a} Then  \exists N \in \mathbb{Z} : \frac{1}{N}< \frac{ \epsilon}{2} .
    Let  k be the largest integer such that  \frac{k}{N} <a . Then  a< \dfrac{k+1}{N} < \dfrac{k+2}{N} <b (There are therefore atleast 2 rationals between any 2 reals).
    But take these rationals and find their mean,  \dfrac{2k+3}{2N} . It can be easily seen that this is distinct from the other 2 rationals. Also can show the mean of any 2 rationals lies in the interval between them if needed. As we have not made any assumptions on the values of the 2 starting rationals, we can take any 2 distinct rationals in our interval and calculate the mean to find a new rational infinite times therefore infinite rationals in interval. (also its clear the mean is a rational as the products of and sums of any integers is an integer).
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    (Original post by EnglishMuon)
    Any feedback on my argument for there being infinitely many rationals in the interval  [a,b] ? (a,b are any reals, b>a)
    Let  \epsilon = \dfrac{1}{b-a} Then  \exists N \in \mathbb{Z} : \frac{1}{N}< \frac{ \epsilon}{2} .
    Let  k be the largest integer such that  \frac{k}{N} <a . Then  a< \dfrac{k+1}{N} < \dfrac{k+2}{N} <b
    Seems okay (although there is a rather slightly shorter shorter proof), but how do you know that \frac{k+2}{N} < b? (I might just be missing something obvious).

    I'd say that \frac{k+2}{N} < \frac{aN + 2}{N} = a + \frac{2}{N} < a + \epsilon where, if \epsilon = b-a, then this inequality would generate a + (b-a) = b and we'd get \frac{k+2}{N} < b as expected, but that's not what \epsilon has been defined as.
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    (Original post by Zacken)
    Seems okay (although there is a rather slightly shorter shorter proof), but how do you know that \frac{k+2}{N} < b? (I might just be missing something obvious).

    I'd say that \frac{k+2}{N} < \frac{aN + 2}{N} = a + \frac{2}{N} < a + \epsilon where, if \epsilon = b-a, then this inequality would generate a + (b-a) = b and we'd get \frac{k+2}{N} < b as expected, but that's not what \epsilon has been defined as.
    yep that makes sense and yea sorry I meant to write my epsilon the other way up as  1/2 (b-a) then it all works out I think, i just wrote upside down
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    Not sure where else to post this:Name:  ImageUploadedByStudent Room1467936421.429411.jpg
Views: 110
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    Definitely want to get involved in this! I'm nowhere near the level you guys are at, but I'm starting Maths and Economics at LSE through distance learning after summer, so want to bridge the gap between what I know and what I'll need to know.
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    While going through N&S I had a question. Are natural numbers positive integers and whole numbers non-negative integers? Or are natural numbers non-negative integers – in which case what are whole numbers? Or are both conventions followed by different people?
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    (Original post by krishdesai7)
    While going through N&S I had a question. Are natural numbers positive integers and whole numbers non-negative integers? Or are natural numbers non-negative integers – in which case what are whole numbers? Or are both conventions followed by different people?
    Natural numbers are the counting numbers, ( the non negative and non-zero integers ) whole numbers are just integers AFAIK


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