Integrating a Radial Function in Hyperspherical Coordinates… But not all of them.

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So I have a function $W$ defined in a hyperdimensional space. I am working with an $n$-dimensional hyperspherical coordinate system where $r$ is the radial part $rin[0,infty)$, $theta_i$ are the Inclinations or Latitudes $theta_i in [0,pi] forall 1<i<n-2$ and $phi$ is the Azimuth $phi in [0,2pi]$.

It turns out that $W(r,theta_1,dots,theta_n-2,phi)$ is a radial function, so points that are only different in their angular coordinates have the same value of $W$.

I want to project $W$ down to 3D, turning it from $W(r,theta_1,dots,theta_n-2,phi)$ to $w(r,theta,phi)$.

What I am trying to do here is integrating $W$ in every possible Inclination coordinate except one in order to obtain $w(r,theta,phi)$, which is a function of a radius, an Azimuth and a single Inclination.

$$
w(r,theta,phi)=int_0^pi int_0^pidotsint_0^pi W(r,theta_1,dots,theta_n-2,phi) prod_j=1^n-3 sin^n-j-1 theta_j ~ mathrmdtheta_1 mathrmd theta_2, dots,mathrmd theta_n-3
$$

This looks totally bonkers, but I was hoping that maybe there is a closed-form for this madness, given the (hyper)spherical symmetry of $W$. Has anyone ever tried to do something like this? Any tips as to why this might not be a good way to go?

integration multivariable-calculus definite-integrals polar-coordinates

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edited Jul 29 at 5:25

asked Jul 20 at 20:02

urquiza

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So I have a function $W$ defined in a hyperdimensional space. I am working with an $n$-dimensional hyperspherical coordinate system where $r$ is the radial part $rin[0,infty)$, $theta_i$ are the Inclinations or Latitudes $theta_i in [0,pi] forall 1<i<n-2$ and $phi$ is the Azimuth $phi in [0,2pi]$.

It turns out that $W(r,theta_1,dots,theta_n-2,phi)$ is a radial function, so points that are only different in their angular coordinates have the same value of $W$.

I want to project $W$ down to 3D, turning it from $W(r,theta_1,dots,theta_n-2,phi)$ to $w(r,theta,phi)$.

What I am trying to do here is integrating $W$ in every possible Inclination coordinate except one in order to obtain $w(r,theta,phi)$, which is a function of a radius, an Azimuth and a single Inclination.

$$
w(r,theta,phi)=int_0^pi int_0^pidotsint_0^pi W(r,theta_1,dots,theta_n-2,phi) prod_j=1^n-3 sin^n-j-1 theta_j ~ mathrmdtheta_1 mathrmd theta_2, dots,mathrmd theta_n-3
$$

This looks totally bonkers, but I was hoping that maybe there is a closed-form for this madness, given the (hyper)spherical symmetry of $W$. Has anyone ever tried to do something like this? Any tips as to why this might not be a good way to go?

integration multivariable-calculus definite-integrals polar-coordinates

share|cite|improve this question

edited Jul 29 at 5:25

asked Jul 20 at 20:02

urquiza

1338

add a comment | 

up vote
0
down vote

favorite

up vote
0
down vote

favorite

So I have a function $W$ defined in a hyperdimensional space. I am working with an $n$-dimensional hyperspherical coordinate system where $r$ is the radial part $rin[0,infty)$, $theta_i$ are the Inclinations or Latitudes $theta_i in [0,pi] forall 1<i<n-2$ and $phi$ is the Azimuth $phi in [0,2pi]$.

It turns out that $W(r,theta_1,dots,theta_n-2,phi)$ is a radial function, so points that are only different in their angular coordinates have the same value of $W$.

I want to project $W$ down to 3D, turning it from $W(r,theta_1,dots,theta_n-2,phi)$ to $w(r,theta,phi)$.

What I am trying to do here is integrating $W$ in every possible Inclination coordinate except one in order to obtain $w(r,theta,phi)$, which is a function of a radius, an Azimuth and a single Inclination.

$$
w(r,theta,phi)=int_0^pi int_0^pidotsint_0^pi W(r,theta_1,dots,theta_n-2,phi) prod_j=1^n-3 sin^n-j-1 theta_j ~ mathrmdtheta_1 mathrmd theta_2, dots,mathrmd theta_n-3
$$

This looks totally bonkers, but I was hoping that maybe there is a closed-form for this madness, given the (hyper)spherical symmetry of $W$. Has anyone ever tried to do something like this? Any tips as to why this might not be a good way to go?

integration multivariable-calculus definite-integrals polar-coordinates

share|cite|improve this question

edited Jul 29 at 5:25

asked Jul 20 at 20:02

urquiza

1338

So I have a function $W$ defined in a hyperdimensional space. I am working with an $n$-dimensional hyperspherical coordinate system where $r$ is the radial part $rin[0,infty)$, $theta_i$ are the Inclinations or Latitudes $theta_i in [0,pi] forall 1<i<n-2$ and $phi$ is the Azimuth $phi in [0,2pi]$.

It turns out that $W(r,theta_1,dots,theta_n-2,phi)$ is a radial function, so points that are only different in their angular coordinates have the same value of $W$.

I want to project $W$ down to 3D, turning it from $W(r,theta_1,dots,theta_n-2,phi)$ to $w(r,theta,phi)$.

What I am trying to do here is integrating $W$ in every possible Inclination coordinate except one in order to obtain $w(r,theta,phi)$, which is a function of a radius, an Azimuth and a single Inclination.

$$
w(r,theta,phi)=int_0^pi int_0^pidotsint_0^pi W(r,theta_1,dots,theta_n-2,phi) prod_j=1^n-3 sin^n-j-1 theta_j ~ mathrmdtheta_1 mathrmd theta_2, dots,mathrmd theta_n-3
$$

This looks totally bonkers, but I was hoping that maybe there is a closed-form for this madness, given the (hyper)spherical symmetry of $W$. Has anyone ever tried to do something like this? Any tips as to why this might not be a good way to go?

integration multivariable-calculus definite-integrals polar-coordinates

share|cite|improve this question

edited Jul 29 at 5:25

asked Jul 20 at 20:02

urquiza

1338

share|cite|improve this question

share|cite|improve this question

edited Jul 29 at 5:25

edited Jul 29 at 5:25

edited Jul 29 at 5:25

asked Jul 20 at 20:02

urquiza

1338

asked Jul 20 at 20:02

urquiza

1338

asked Jul 20 at 20:02

urquiza

1338

1338

add a comment | 

add a comment | 

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