Having issues understanding the notion of 2D volume in Measure Theory.
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So the exercise I'm trying to solve goes like this:
P â $R^2$ is the interior of a parallelogram defined by its' corners coordinates:
(-2,1), (-1,3), (-3,3), (-4,1)
I need to find out $vol_2 (P)$
Now I found somewhere in a book that if the set I'm looking at is
$P=[a_1,b_1] times [a_2,b_2] times ... times [a_n,b_n]$ â $R^n$
then
$vol_n (P)$=$|b_1-a_1| |b_1-a_1|...|b_n-a_n|$
And I believe this finding may prove to be useful in what I need to solve, but I fail to identify the terms $[a_1,b_1]$ and $[a_2,b_2]$ in my scenario. Also, I don't really understand the concept of having a volume if this is a 2 dimensional figure.
integration measure-theory lebesgue-measure
 |Â
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up vote
0
down vote
favorite
So the exercise I'm trying to solve goes like this:
P â $R^2$ is the interior of a parallelogram defined by its' corners coordinates:
(-2,1), (-1,3), (-3,3), (-4,1)
I need to find out $vol_2 (P)$
Now I found somewhere in a book that if the set I'm looking at is
$P=[a_1,b_1] times [a_2,b_2] times ... times [a_n,b_n]$ â $R^n$
then
$vol_n (P)$=$|b_1-a_1| |b_1-a_1|...|b_n-a_n|$
And I believe this finding may prove to be useful in what I need to solve, but I fail to identify the terms $[a_1,b_1]$ and $[a_2,b_2]$ in my scenario. Also, I don't really understand the concept of having a volume if this is a 2 dimensional figure.
integration measure-theory lebesgue-measure
Your formula won't work, because it's not a 2 dimensional cube. Also, specify your question. What options do you have to evaluate the area. You can obviously just use elementary school formulas, but I guess that's a no?
â Rumpelstiltskin
Aug 8 at 21:34
The author's note was that the formula is for any n-dimensional parallelepiped. Yeah I'm not supposed to use elementary school geometry, but measure theory and the likes of Lebesgue integration.
â Kira HD
Aug 8 at 21:37
I guess you can just integrate $1$ over your set? Would that do? Your formula doesn't exactly work here, it's not of the form like $P$
â Rumpelstiltskin
Aug 8 at 21:43
I have no idea sadly..
â Kira HD
Aug 8 at 21:44
Did you cover double integrals?
â Rumpelstiltskin
Aug 8 at 21:44
 |Â
show 3 more comments
up vote
0
down vote
favorite
up vote
0
down vote
favorite
So the exercise I'm trying to solve goes like this:
P â $R^2$ is the interior of a parallelogram defined by its' corners coordinates:
(-2,1), (-1,3), (-3,3), (-4,1)
I need to find out $vol_2 (P)$
Now I found somewhere in a book that if the set I'm looking at is
$P=[a_1,b_1] times [a_2,b_2] times ... times [a_n,b_n]$ â $R^n$
then
$vol_n (P)$=$|b_1-a_1| |b_1-a_1|...|b_n-a_n|$
And I believe this finding may prove to be useful in what I need to solve, but I fail to identify the terms $[a_1,b_1]$ and $[a_2,b_2]$ in my scenario. Also, I don't really understand the concept of having a volume if this is a 2 dimensional figure.
integration measure-theory lebesgue-measure
So the exercise I'm trying to solve goes like this:
P â $R^2$ is the interior of a parallelogram defined by its' corners coordinates:
(-2,1), (-1,3), (-3,3), (-4,1)
I need to find out $vol_2 (P)$
Now I found somewhere in a book that if the set I'm looking at is
$P=[a_1,b_1] times [a_2,b_2] times ... times [a_n,b_n]$ â $R^n$
then
$vol_n (P)$=$|b_1-a_1| |b_1-a_1|...|b_n-a_n|$
And I believe this finding may prove to be useful in what I need to solve, but I fail to identify the terms $[a_1,b_1]$ and $[a_2,b_2]$ in my scenario. Also, I don't really understand the concept of having a volume if this is a 2 dimensional figure.
integration measure-theory lebesgue-measure
asked Aug 8 at 21:32
Kira HD
143
143
Your formula won't work, because it's not a 2 dimensional cube. Also, specify your question. What options do you have to evaluate the area. You can obviously just use elementary school formulas, but I guess that's a no?
â Rumpelstiltskin
Aug 8 at 21:34
The author's note was that the formula is for any n-dimensional parallelepiped. Yeah I'm not supposed to use elementary school geometry, but measure theory and the likes of Lebesgue integration.
â Kira HD
Aug 8 at 21:37
I guess you can just integrate $1$ over your set? Would that do? Your formula doesn't exactly work here, it's not of the form like $P$
â Rumpelstiltskin
Aug 8 at 21:43
I have no idea sadly..
â Kira HD
Aug 8 at 21:44
Did you cover double integrals?
â Rumpelstiltskin
Aug 8 at 21:44
 |Â
show 3 more comments
Your formula won't work, because it's not a 2 dimensional cube. Also, specify your question. What options do you have to evaluate the area. You can obviously just use elementary school formulas, but I guess that's a no?
â Rumpelstiltskin
Aug 8 at 21:34
The author's note was that the formula is for any n-dimensional parallelepiped. Yeah I'm not supposed to use elementary school geometry, but measure theory and the likes of Lebesgue integration.
â Kira HD
Aug 8 at 21:37
I guess you can just integrate $1$ over your set? Would that do? Your formula doesn't exactly work here, it's not of the form like $P$
â Rumpelstiltskin
Aug 8 at 21:43
I have no idea sadly..
â Kira HD
Aug 8 at 21:44
Did you cover double integrals?
â Rumpelstiltskin
Aug 8 at 21:44
Your formula won't work, because it's not a 2 dimensional cube. Also, specify your question. What options do you have to evaluate the area. You can obviously just use elementary school formulas, but I guess that's a no?
â Rumpelstiltskin
Aug 8 at 21:34
Your formula won't work, because it's not a 2 dimensional cube. Also, specify your question. What options do you have to evaluate the area. You can obviously just use elementary school formulas, but I guess that's a no?
â Rumpelstiltskin
Aug 8 at 21:34
The author's note was that the formula is for any n-dimensional parallelepiped. Yeah I'm not supposed to use elementary school geometry, but measure theory and the likes of Lebesgue integration.
â Kira HD
Aug 8 at 21:37
The author's note was that the formula is for any n-dimensional parallelepiped. Yeah I'm not supposed to use elementary school geometry, but measure theory and the likes of Lebesgue integration.
â Kira HD
Aug 8 at 21:37
I guess you can just integrate $1$ over your set? Would that do? Your formula doesn't exactly work here, it's not of the form like $P$
â Rumpelstiltskin
Aug 8 at 21:43
I guess you can just integrate $1$ over your set? Would that do? Your formula doesn't exactly work here, it's not of the form like $P$
â Rumpelstiltskin
Aug 8 at 21:43
I have no idea sadly..
â Kira HD
Aug 8 at 21:44
I have no idea sadly..
â Kira HD
Aug 8 at 21:44
Did you cover double integrals?
â Rumpelstiltskin
Aug 8 at 21:44
Did you cover double integrals?
â Rumpelstiltskin
Aug 8 at 21:44
 |Â
show 3 more comments
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Your formula won't work, because it's not a 2 dimensional cube. Also, specify your question. What options do you have to evaluate the area. You can obviously just use elementary school formulas, but I guess that's a no?
â Rumpelstiltskin
Aug 8 at 21:34
The author's note was that the formula is for any n-dimensional parallelepiped. Yeah I'm not supposed to use elementary school geometry, but measure theory and the likes of Lebesgue integration.
â Kira HD
Aug 8 at 21:37
I guess you can just integrate $1$ over your set? Would that do? Your formula doesn't exactly work here, it's not of the form like $P$
â Rumpelstiltskin
Aug 8 at 21:43
I have no idea sadly..
â Kira HD
Aug 8 at 21:44
Did you cover double integrals?
â Rumpelstiltskin
Aug 8 at 21:44