Calculate the integral of the following function over the sphere in $mathbb{R}^4$
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Calculate the integral of $f(x,y,w,z) = |y||z|^2$ over $S={(x,y,w,z)|x^2+y^2+z^2+w^2=1}$, i.e. calculate: $int_S|y||z|^2dsigma_3$.
My attempt -
I'll define a parameterization as follows - $phi(r,alpha,beta)=(rcos(alpha),rsin(alpha),rcos(beta),rsin(beta))$.
Using the following formula: $int_Mf dS = intintint fcirc phi sqrt{det(D_phi*D_phi^T)}drdalpha dbeta$
Putting it all together I get that $int_S|y||z|^2dsigma_3 = int_0^{2pi}int_0^{2pi}int_0^1|rsin(alpha)|r^2sin(beta)^2sqrt{2}r^2drdalpha dbeta = frac{sqrt{2}}{6}piint_0^{2pi}|sin(alpha)|dalpha = frac{sqrt{2}}{6}pi(int_0^pi sin(alpha) dalpha - int_pi^{2pi} sin(alpha) dalpha) = frac{4sqrt{2}}{6}pi = frac{2sqrt{2}}{3}pi$
Is my calculation correct?
And if not, what is the right way to calculate it?
calculus integration multivariable-calculus manifolds spheres
asked Dec 15 '18 at 16:06
ChikChakChikChak
754418
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show 1 more comment
$begingroup$
Calculate the integral of $f(x,y,w,z) = |y||z|^2$ over $S={(x,y,w,z)|x^2+y^2+z^2+w^2=1}$, i.e. calculate: $int_S|y||z|^2dsigma_3$.
My attempt -
I'll define a parameterization as follows - $phi(r,alpha,beta)=(rcos(alpha),rsin(alpha),rcos(beta),rsin(beta))$.
Using the following formula: $int_Mf dS = intintint fcirc phi sqrt{det(D_phi*D_phi^T)}drdalpha dbeta$
Putting it all together I get that $int_S|y||z|^2dsigma_3 = int_0^{2pi}int_0^{2pi}int_0^1|rsin(alpha)|r^2sin(beta)^2sqrt{2}r^2drdalpha dbeta = frac{sqrt{2}}{6}piint_0^{2pi}|sin(alpha)|dalpha = frac{sqrt{2}}{6}pi(int_0^pi sin(alpha) dalpha - int_pi^{2pi} sin(alpha) dalpha) = frac{4sqrt{2}}{6}pi = frac{2sqrt{2}}{3}pi$
Is my calculation correct?
And if not, what is the right way to calculate it?
calculus integration multivariable-calculus manifolds spheres
asked Dec 15 '18 at 16:06
ChikChakChikChak
754418
$endgroup$
$begingroup$
Are you asking how to do the single variable integral that you've arrived at?
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– jgon
Dec 15 '18 at 16:11
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@jgon Edited... Is my final result correct?
$endgroup$
– ChikChak
Dec 15 '18 at 16:15
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Well the final result certainly can't be zero, so no, and you haven't done the single variable integral correctly, since that integral also must be positive.
$endgroup$
– jgon
Dec 15 '18 at 16:16
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@jgon Though, is the way up to the single variable integral correct?
$endgroup$
– ChikChak
Dec 15 '18 at 16:19
$begingroup$
Not sure, I didn't want to compute the determinant of the matrix.
$endgroup$
– jgon
Dec 15 '18 at 16:20
|
show 1 more comment
$begingroup$
Calculate the integral of $f(x,y,w,z) = |y||z|^2$ over $S={(x,y,w,z)|x^2+y^2+z^2+w^2=1}$, i.e. calculate: $int_S|y||z|^2dsigma_3$.
My attempt -
I'll define a parameterization as follows - $phi(r,alpha,beta)=(rcos(alpha),rsin(alpha),rcos(beta),rsin(beta))$.
Using the following formula: $int_Mf dS = intintint fcirc phi sqrt{det(D_phi*D_phi^T)}drdalpha dbeta$
Putting it all together I get that $int_S|y||z|^2dsigma_3 = int_0^{2pi}int_0^{2pi}int_0^1|rsin(alpha)|r^2sin(beta)^2sqrt{2}r^2drdalpha dbeta = frac{sqrt{2}}{6}piint_0^{2pi}|sin(alpha)|dalpha = frac{sqrt{2}}{6}pi(int_0^pi sin(alpha) dalpha - int_pi^{2pi} sin(alpha) dalpha) = frac{4sqrt{2}}{6}pi = frac{2sqrt{2}}{3}pi$
Is my calculation correct?
And if not, what is the right way to calculate it?
calculus integration multivariable-calculus manifolds spheres
asked Dec 15 '18 at 16:06
ChikChakChikChak
754418
$endgroup$
Calculate the integral of $f(x,y,w,z) = |y||z|^2$ over $S={(x,y,w,z)|x^2+y^2+z^2+w^2=1}$, i.e. calculate: $int_S|y||z|^2dsigma_3$.
My attempt -
I'll define a parameterization as follows - $phi(r,alpha,beta)=(rcos(alpha),rsin(alpha),rcos(beta),rsin(beta))$.
Using the following formula: $int_Mf dS = intintint fcirc phi sqrt{det(D_phi*D_phi^T)}drdalpha dbeta$
Putting it all together I get that $int_S|y||z|^2dsigma_3 = int_0^{2pi}int_0^{2pi}int_0^1|rsin(alpha)|r^2sin(beta)^2sqrt{2}r^2drdalpha dbeta = frac{sqrt{2}}{6}piint_0^{2pi}|sin(alpha)|dalpha = frac{sqrt{2}}{6}pi(int_0^pi sin(alpha) dalpha - int_pi^{2pi} sin(alpha) dalpha) = frac{4sqrt{2}}{6}pi = frac{2sqrt{2}}{3}pi$
Is my calculation correct?
And if not, what is the right way to calculate it?
calculus integration multivariable-calculus manifolds spheres
calculus integration multivariable-calculus manifolds spheres
asked Dec 15 '18 at 16:06
ChikChakChikChak
754418
asked Dec 15 '18 at 16:06
ChikChakChikChak
754418
edited Dec 15 '18 at 17:05
ChikChak
asked Dec 15 '18 at 16:06
ChikChakChikChak
754418
asked Dec 15 '18 at 16:06
ChikChakChikChak
754418
asked Dec 15 '18 at 16:06
ChikChakChikChak
754418
754418
$begingroup$
Are you asking how to do the single variable integral that you've arrived at?
$endgroup$
– jgon
Dec 15 '18 at 16:11
$begingroup$
@jgon Edited... Is my final result correct?
$endgroup$
– ChikChak
Dec 15 '18 at 16:15
$begingroup$
Well the final result certainly can't be zero, so no, and you haven't done the single variable integral correctly, since that integral also must be positive.
$endgroup$
– jgon
Dec 15 '18 at 16:16
$begingroup$
@jgon Though, is the way up to the single variable integral correct?
$endgroup$
– ChikChak
Dec 15 '18 at 16:19
$begingroup$
Not sure, I didn't want to compute the determinant of the matrix.
$endgroup$
– jgon
Dec 15 '18 at 16:20
|
show 1 more comment
$begingroup$
Are you asking how to do the single variable integral that you've arrived at?
$endgroup$
– jgon
Dec 15 '18 at 16:11
$begingroup$
@jgon Edited... Is my final result correct?
$endgroup$
– ChikChak
Dec 15 '18 at 16:15
$begingroup$
Well the final result certainly can't be zero, so no, and you haven't done the single variable integral correctly, since that integral also must be positive.
$endgroup$
– jgon
Dec 15 '18 at 16:16
$begingroup$
@jgon Though, is the way up to the single variable integral correct?
$endgroup$
– ChikChak
Dec 15 '18 at 16:19
$begingroup$
Not sure, I didn't want to compute the determinant of the matrix.
$endgroup$
– jgon
Dec 15 '18 at 16:20
$begingroup$
Are you asking how to do the single variable integral that you've arrived at?
$endgroup$
– jgon
Dec 15 '18 at 16:11
$begingroup$
Are you asking how to do the single variable integral that you've arrived at?
$endgroup$
– jgon
Dec 15 '18 at 16:11
$begingroup$
@jgon Edited... Is my final result correct?
$endgroup$
– ChikChak
Dec 15 '18 at 16:15
$begingroup$
@jgon Edited... Is my final result correct?
$endgroup$
– ChikChak
Dec 15 '18 at 16:15
$begingroup$
Well the final result certainly can't be zero, so no, and you haven't done the single variable integral correctly, since that integral also must be positive.
$endgroup$
– jgon
Dec 15 '18 at 16:16
$begingroup$
Well the final result certainly can't be zero, so no, and you haven't done the single variable integral correctly, since that integral also must be positive.
$endgroup$
– jgon
Dec 15 '18 at 16:16
$begingroup$
@jgon Though, is the way up to the single variable integral correct?
$endgroup$
– ChikChak
Dec 15 '18 at 16:19
$begingroup$
@jgon Though, is the way up to the single variable integral correct?
$endgroup$
– ChikChak
Dec 15 '18 at 16:19
$begingroup$
Not sure, I didn't want to compute the determinant of the matrix.
$endgroup$
– jgon
Dec 15 '18 at 16:20
$begingroup$
Not sure, I didn't want to compute the determinant of the matrix.
$endgroup$
– jgon
Dec 15 '18 at 16:20
|
show 1 more comment
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$begingroup$
Are you asking how to do the single variable integral that you've arrived at?
$endgroup$
– jgon
Dec 15 '18 at 16:11
$begingroup$
@jgon Edited... Is my final result correct?
$endgroup$
– ChikChak
Dec 15 '18 at 16:15
$begingroup$
Well the final result certainly can't be zero, so no, and you haven't done the single variable integral correctly, since that integral also must be positive.
$endgroup$
– jgon
Dec 15 '18 at 16:16
$begingroup$
@jgon Though, is the way up to the single variable integral correct?
$endgroup$
– ChikChak
Dec 15 '18 at 16:19
$begingroup$
Not sure, I didn't want to compute the determinant of the matrix.
$endgroup$
– jgon
Dec 15 '18 at 16:20