a question on the volume of a set
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Can anybody give me a hint on this?
Let $vin mathbb{R}^n,quad r,R,C$ positive constants .
We would like to calculate the volume of the set $$A={win mathbb{R}^n : rleq |w| leq R, |w-v/2|leq C }$$
It seems like the intersection of an annulus and a ball (both in the n-dimensional sense) but i can't write down an integral properly.
integration multivariable-calculus volume
asked Dec 27 '18 at 19:34
PmorphyPmorphy
1097
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add a comment |
$begingroup$
Can anybody give me a hint on this?
Let $vin mathbb{R}^n,quad r,R,C$ positive constants .
We would like to calculate the volume of the set $$A={win mathbb{R}^n : rleq |w| leq R, |w-v/2|leq C }$$
It seems like the intersection of an annulus and a ball (both in the n-dimensional sense) but i can't write down an integral properly.
integration multivariable-calculus volume
asked Dec 27 '18 at 19:34
PmorphyPmorphy
1097
$endgroup$
$begingroup$
You need to figure out the volume of intersection between two hypersphere and substract.... $$mu(A) = muleft((bar{B}_R(0) setminus B_r(0))cap bar{B}_C(frac{v}{2})right) = muleft(bar{B}_R(0) cap bar{B}_C(frac{v}{2})right) -muleft(B_r(0) cap bar{B}_C(frac{v}{2})right) $$
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– achille hui
Dec 27 '18 at 19:46
$begingroup$
To compute the volume of intersection between two hypershere, see answer of this and the wiki entry on hyperspherical cap.
$endgroup$
– achille hui
Dec 27 '18 at 19:54
add a comment |
$begingroup$
Can anybody give me a hint on this?
Let $vin mathbb{R}^n,quad r,R,C$ positive constants .
We would like to calculate the volume of the set $$A={win mathbb{R}^n : rleq |w| leq R, |w-v/2|leq C }$$
It seems like the intersection of an annulus and a ball (both in the n-dimensional sense) but i can't write down an integral properly.
integration multivariable-calculus volume
asked Dec 27 '18 at 19:34
PmorphyPmorphy
1097
$endgroup$
Can anybody give me a hint on this?
Let $vin mathbb{R}^n,quad r,R,C$ positive constants .
We would like to calculate the volume of the set $$A={win mathbb{R}^n : rleq |w| leq R, |w-v/2|leq C }$$
It seems like the intersection of an annulus and a ball (both in the n-dimensional sense) but i can't write down an integral properly.
integration multivariable-calculus volume
integration multivariable-calculus volume
asked Dec 27 '18 at 19:34
PmorphyPmorphy
1097
asked Dec 27 '18 at 19:34
PmorphyPmorphy
1097
asked Dec 27 '18 at 19:34
PmorphyPmorphy
1097
asked Dec 27 '18 at 19:34
PmorphyPmorphy
1097
asked Dec 27 '18 at 19:34
PmorphyPmorphy
1097
1097
$begingroup$
You need to figure out the volume of intersection between two hypersphere and substract.... $$mu(A) = muleft((bar{B}_R(0) setminus B_r(0))cap bar{B}_C(frac{v}{2})right) = muleft(bar{B}_R(0) cap bar{B}_C(frac{v}{2})right) -muleft(B_r(0) cap bar{B}_C(frac{v}{2})right) $$
$endgroup$
– achille hui
Dec 27 '18 at 19:46
$begingroup$
To compute the volume of intersection between two hypershere, see answer of this and the wiki entry on hyperspherical cap.
$endgroup$
– achille hui
Dec 27 '18 at 19:54
add a comment |
$begingroup$
You need to figure out the volume of intersection between two hypersphere and substract.... $$mu(A) = muleft((bar{B}_R(0) setminus B_r(0))cap bar{B}_C(frac{v}{2})right) = muleft(bar{B}_R(0) cap bar{B}_C(frac{v}{2})right) -muleft(B_r(0) cap bar{B}_C(frac{v}{2})right) $$
$endgroup$
– achille hui
Dec 27 '18 at 19:46
$begingroup$
To compute the volume of intersection between two hypershere, see answer of this and the wiki entry on hyperspherical cap.
$endgroup$
– achille hui
Dec 27 '18 at 19:54
$begingroup$
You need to figure out the volume of intersection between two hypersphere and substract.... $$mu(A) = muleft((bar{B}_R(0) setminus B_r(0))cap bar{B}_C(frac{v}{2})right) = muleft(bar{B}_R(0) cap bar{B}_C(frac{v}{2})right) -muleft(B_r(0) cap bar{B}_C(frac{v}{2})right) $$
$endgroup$
– achille hui
Dec 27 '18 at 19:46
$begingroup$
You need to figure out the volume of intersection between two hypersphere and substract.... $$mu(A) = muleft((bar{B}_R(0) setminus B_r(0))cap bar{B}_C(frac{v}{2})right) = muleft(bar{B}_R(0) cap bar{B}_C(frac{v}{2})right) -muleft(B_r(0) cap bar{B}_C(frac{v}{2})right) $$
$endgroup$
– achille hui
Dec 27 '18 at 19:46
$begingroup$
To compute the volume of intersection between two hypershere, see answer of this and the wiki entry on hyperspherical cap.
$endgroup$
– achille hui
Dec 27 '18 at 19:54
$begingroup$
To compute the volume of intersection between two hypershere, see answer of this and the wiki entry on hyperspherical cap.
$endgroup$
– achille hui
Dec 27 '18 at 19:54
add a comment |
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$begingroup$
You need to figure out the volume of intersection between two hypersphere and substract.... $$mu(A) = muleft((bar{B}_R(0) setminus B_r(0))cap bar{B}_C(frac{v}{2})right) = muleft(bar{B}_R(0) cap bar{B}_C(frac{v}{2})right) -muleft(B_r(0) cap bar{B}_C(frac{v}{2})right) $$
$endgroup$
– achille hui
Dec 27 '18 at 19:46
$begingroup$
To compute the volume of intersection between two hypershere, see answer of this and the wiki entry on hyperspherical cap.
$endgroup$
– achille hui
Dec 27 '18 at 19:54