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anonymous

  • one year ago

"Since integral curves can’t cross an integral curve itself is both an upper and a lower fence" Can anyone explain this for me a little more. I'm not sure if i'm getting it right. I kind of understand this "An integral curve is also a fence of both types" The quote is a Remark in this http://ocw.mit.edu/courses/mathematics/18-03sc-differential-equations-fall-2011/unit-i-first-order-differential-equations/geometric-methods/MIT18_03SCF11_s2_5text.pdf

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  1. ganeshie8
    • one year ago
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    |dw:1437290124325:dw|

  2. ganeshie8
    • one year ago
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    Because to go outside the fence requires to cross the fence; at the point of crossing, there will be two different slopes. But the direction field allows you to have only one slope at a point.

  3. anonymous
    • one year ago
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    Thanks for answering, i knew that. I just got confused by that specific remark, like how an integral curve itself could be both an upper and a lower fence, arent the integral curve suppose to be between them (if conditions says so)

  4. ganeshie8
    • one year ago
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    so the solution curves inside the fence stay inside forever the solution curves outside the fence stay outside forever |dw:1437290828331:dw|

  5. anonymous
    • one year ago
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    Oh. I think i get it now. So the red integral curve is algo being a lower fence for the purple integral curve.

  6. anonymous
    • one year ago
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    Btw, i know that if an integral curve is inside a fence it never gets out, i'm just wondering about that specif remark saying an integral curve is also a fence

  7. ganeshie8
    • one year ago
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    yes that follows from the fact that the integral curves cannot touch each other

  8. anonymous
    • one year ago
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    Thanks, i think i understood

  9. ganeshie8
    • one year ago
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    the fence shown in the diagram is indeed an integral curve

  10. ganeshie8
    • one year ago
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    don't get tricked by the word "fence"

  11. ganeshie8
    • one year ago
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    |dw:1437291167999:dw|

  12. anonymous
    • one year ago
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    Sorry you lost me again

  13. anonymous
    • one year ago
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    i meant i'm lost again xD

  14. anonymous
    • one year ago
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    straight black lines are also integral curves, why?

  15. ganeshie8
    • one year ago
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    Yes, they are integral curves, it is given.

  16. anonymous
    • one year ago
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    Is it? it just says those lines are fences and that any integral curve between them will remaing there forever. How comes those fences are now integral curves.

  17. ganeshie8
    • one year ago
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    If they are not integral curves, then there is nothing stopping to have a solution curve like this : |dw:1437291488143:dw|