A question about green's theorem... Watch

Artus
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I am a bit confused about what a "simple curve" is...

On page 1049 of my textbook (Multivariable Calculus: Early Transcendentals by James Stewart), it defines a "simple curve" as "...a curve that doesn't intersect itself anywhere between its endpoints."But on page 1059, when it gives an example about green's theorem...it says that a the area between two semicircles is not simple...but the curve surrounding the area between the two semicircles does not intersect itself more than once...then why does my textbook say that the region is not simple?

This is the example...

"Evaluate \int y^2 dx + 3xy dy (there is a small circle on the integral sign but I couldn't type that), where C is the boundary of hte semiannular region D in the upper half-plane between the circles x^2 + y^2 = 1 and x^2 + y^2 = 4.

Solution: Notice that although D is not simple, the y-axis divides it into two simple regions..."


So my question is: why is the region D not considered "simple"?

Thank you in advance...
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iluvmaths
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if u dont mind me asking, are u in university? and what year
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Artus
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(Original post by iluvmaths)
if u dont mind me asking, are u in university? and what year
yes...sophomore...why?
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iluvmaths
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becasue, i have only read the calculus version by stewart yet, not fully but considering ur a sophomore, i dont think anyone on tsr, has actually done Multivariable calculus considering this is only a students forum... probably, dr.D might help.. but im not sure.. have u read the calculus version btw?
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iluvmaths
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the only possible answer i can give u is that maybe, he doesnt define the semicircle as a simpl curve because it intersect at two endpoints.. hence it cannot have a simple area
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Artus
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(Original post by iluvmaths)
becasue, i have only read the calculus version by stewart yet, not fully but considering ur a sophomore, i dont think anyone on tsr, has actually done Multivariable calculus considering this is only a students forum... probably, dr.D might help.. but im not sure.. have u read the calculus version btw?
What do you mean by "calculus version"?
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iluvmaths
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thats the version which has calculus written on the front and and integral sign.. my fav book.. its the metric international edition.
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Artus
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(Original post by iluvmaths)
the only possible answer i can give u is that maybe, he doesnt define the semicircle as a simpl curve because it intersect at two endpoints.. hence it cannot have a simple area
But the semicircle doesn't interstect between the two endpoints...
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suneilr
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(Original post by Artus)
...
Could it be something to do with the fact that for the upper half plane semicircle, every y value is mapped to by two x values, but the y-axis separates it into two section where it is injective?
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Artus
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(Original post by iluvmaths)
thats the version which has calculus written on the front and and integral sign.. my fav book.. its the metric international edition.
I don't know...the textbook I'm using now also has an integral sign and the word "Calculus"on the cover...
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Artus
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(Original post by suneilr)
Could it be something to do with the fact that for the upper half plane semicircle, every y value is mapped to by two x values, but the y-axis separates it into two section where it is injective?
...but the definition of a "simple curve" doesn't say anything about the mapping of x and y values...it clearly states that a simple curve is a curve that does not intersect between the endpoints.
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iluvmaths
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think of the 2 superposed curves as one curve and then they intersect hence not simple
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suneilr
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(Original post by Artus)
...but the definition of a "simple curve" doesn't say anything about the mapping of x and y values...it clearly states that a simple curve is a curve that does not intersect between the endpoints.
Well it says that D is simple, and D is a region not a curve. Is there a definition of a simple region? Or is just defined in terms of a region bounded by simple curves?
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rowzee
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A simple curve and a simple region are distinct concepts. It is correct that a simple curve does not "intersect itself'' (i.e. is injective). On the other hand, a simple region, say in 2D, is one which is bounded by two straight lines (either top/bottom or right/left). See picture here for a vertically simple (x-simple) region.
These are of the form
- vertically or x-simple
- horizontally or y-simple.

In your semi-annulus example, I am not entirely sure what they mean by a simple region, since the whole region is clearly x-simple (vertically), but not y-simple. On the other hand, the two regions you get separated by the y-axis are both x and y simple by symmetry, so maybe (probably) this is what is meant by a ''simple region''. In any case, there should be a definition of a simple region in the book.
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Artus
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(Original post by rowzee)
A simple curve and a simple region are distinct concepts. It is correct that a simple curve does not "intersect itself'' (i.e. is injective). On the other hand, a simple region, say in 2D, is one which is bounded by two straight lines (either top/bottom or right/left). See picture here for a vertically simple (x-simple) region.
These are of the form
- vertically or x-simple
- horizontally or y-simple.

In your semi-annulus example, I am not entirely sure what they mean by a simple region, since the whole region is clearly x-simple (vertically), but not y-simple. On the other hand, the two regions you get separated by the y-axis are both x and y simple by symmetry, so maybe (probably) this is what is meant by a ''simple region''. In any case, there should be a definition of a simple region in the book.

Thanks for explaining...I just saw the definition of simple regions on page 1056..."Green's Theorem is not easy to prove in general, but we cangive a proof for the special case where the region is both of type I and II (see section 15.3). Let's call such regions simple regions."...but it still doesn't clarify why the region D is simple...so the way they define it is exactly how you did...
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Vaduz
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There are two basic types of regions in the xy-coordinate plane:Type I Regions and Type II Regions.You can find these in your textbook.
http://brookscole.cengage.com/math_d...grals/p00a.pdf
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