"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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"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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|dw:1437290124325:dw|
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.
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)

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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|
Oh. I think i get it now. So the red integral curve is algo being a lower fence for the purple integral curve.
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
yes that follows from the fact that the integral curves cannot touch each other
Thanks, i think i understood
the fence shown in the diagram is indeed an integral curve
don't get tricked by the word "fence"
|dw:1437291167999:dw|
Sorry you lost me again
i meant i'm lost again xD
straight black lines are also integral curves, why?
Yes, they are integral curves, it is given.
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.
If they are not integral curves, then there is nothing stopping to have a solution curve like this : |dw:1437291488143:dw|