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Hi, I'm having some trouble understanding what's going on when integrating
the region M on page 10 of http://www.math.boun.edu.tr/instructors/ozturk/eskiders/fall04math488/bachman.pdf" , It may just be the language.
ƒ : ℝ² → ℝ defined by (x,y) ↦ z = ƒ(x,y) = y² is the function we're
integrating over the top half of the unit circle.
1: I think what he's trying to communicate in this derivation is the
standard double integral, \int \ \int_M \ f(x,y) \ dA \ = \ \int \ \int_M \ y^2 \ dy \ dx.
Is that correct? You'll notice he jumps straight into his paramaterization
but would what I've just done here be right?
2: If so then would the bounds on the integral become:
\int \ \int_M \ f(x,y) \ dA \ = \ \int_{-1}^1 \ \int_0^{( \sqrt{1 - x^2})} \ y^2 \ dy \ dx ?
3: If that is correct then I think it would explain why the author chose to
set up a paramaterization of the region M. When he goes on to show that
the unit circle can be paramaterized in different ways it reduces a double
integral to a single integral & is just easier. Is that why?
4: I've never seen anyone paramaterize double integrals in the way he
does, could you recommend some reading material that explains what he
is doing as I can't seem to find any myself.
I have more questions, mainly to do with pages 11-14 where, I think, he is
deriving differential forms (in my meagre estimation) but I'll hold off for
now, thanks for any assistance!
the region M on page 10 of http://www.math.boun.edu.tr/instructors/ozturk/eskiders/fall04math488/bachman.pdf" , It may just be the language.
ƒ : ℝ² → ℝ defined by (x,y) ↦ z = ƒ(x,y) = y² is the function we're
integrating over the top half of the unit circle.
1: I think what he's trying to communicate in this derivation is the
standard double integral, \int \ \int_M \ f(x,y) \ dA \ = \ \int \ \int_M \ y^2 \ dy \ dx.
Is that correct? You'll notice he jumps straight into his paramaterization
but would what I've just done here be right?
2: If so then would the bounds on the integral become:
\int \ \int_M \ f(x,y) \ dA \ = \ \int_{-1}^1 \ \int_0^{( \sqrt{1 - x^2})} \ y^2 \ dy \ dx ?
3: If that is correct then I think it would explain why the author chose to
set up a paramaterization of the region M. When he goes on to show that
the unit circle can be paramaterized in different ways it reduces a double
integral to a single integral & is just easier. Is that why?
4: I've never seen anyone paramaterize double integrals in the way he
does, could you recommend some reading material that explains what he
is doing as I can't seem to find any myself.
I have more questions, mainly to do with pages 11-14 where, I think, he is
deriving differential forms (in my meagre estimation) but I'll hold off for
now, thanks for any assistance!
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