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What is a cookbook or cheat book for this types of multi-variable integrals?
Clash Royale CLAN TAG#URR8PPP
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I saw somewhere the following integral:
$$ int_mathbbR^d Vert xVert^3 e^-aVert xVert^2dx=a^-fracM+32pi^fracm2fracGamma(fracm+32)Gamma(fracm2)$$
What is a source which lists this types of integrals? A kind of cookbook, which we can look up for this types of multivariable integrals?
If there isn't any handbook, how to solve it?
I searched the Internet, but didn't find any thing.
calculus multivariable-calculus
asked Jul 18 at 13:50
user85361
331215
 |Â
show 5 more comments
up vote
1
down vote
favorite
I saw somewhere the following integral:
$$ int_mathbbR^d Vert xVert^3 e^-aVert xVert^2dx=a^-fracM+32pi^fracm2fracGamma(fracm+32)Gamma(fracm2)$$
What is a source which lists this types of integrals? A kind of cookbook, which we can look up for this types of multivariable integrals?
If there isn't any handbook, how to solve it?
I searched the Internet, but didn't find any thing.
calculus multivariable-calculus
asked Jul 18 at 13:50
user85361
331215
Try Paul J Nahin's book ... Inside Interesting Integrals ... to learn amazing techniques to solve these sorts of integrals
– Bruce
Jul 18 at 13:58
1
Try Wolfram alpha
– Lorenzo
Jul 18 at 13:59
@Bruce, I didn't find any multiple integral$n>2$ in this book. It is a very good book, but I want a cookbook just for reference.
– user85361
Jul 18 at 14:21
1
This is the closet I could get it wolframalpha.com/input/… more woflram-fu might be able to get you further. E.g. you might want to convert it to polar coordinates first, and have wolfram evaluate the 1 variable integral.
– Lorenzo
Jul 18 at 15:18
1
@user85361 Your integral is radially symmetric, so you can compute the integral in the radial direction separately from the rest. E.g. as in here: en.wikipedia.org/wiki/…
– Lorenzo
Jul 19 at 5:47
 |Â
show 5 more comments
up vote
1
down vote
favorite
up vote
1
down vote
favorite
I saw somewhere the following integral:
$$ int_mathbbR^d Vert xVert^3 e^-aVert xVert^2dx=a^-fracM+32pi^fracm2fracGamma(fracm+32)Gamma(fracm2)$$
What is a source which lists this types of integrals? A kind of cookbook, which we can look up for this types of multivariable integrals?
If there isn't any handbook, how to solve it?
I searched the Internet, but didn't find any thing.
calculus multivariable-calculus
asked Jul 18 at 13:50
user85361
331215
I saw somewhere the following integral:
$$ int_mathbbR^d Vert xVert^3 e^-aVert xVert^2dx=a^-fracM+32pi^fracm2fracGamma(fracm+32)Gamma(fracm2)$$
What is a source which lists this types of integrals? A kind of cookbook, which we can look up for this types of multivariable integrals?
If there isn't any handbook, how to solve it?
I searched the Internet, but didn't find any thing.
calculus multivariable-calculus
asked Jul 18 at 13:50
user85361
331215
edited Jul 21 at 13:05
asked Jul 18 at 13:50
user85361
331215
asked Jul 18 at 13:50
user85361
331215
asked Jul 18 at 13:50
user85361
331215
331215
Try Paul J Nahin's book ... Inside Interesting Integrals ... to learn amazing techniques to solve these sorts of integrals
– Bruce
Jul 18 at 13:58
1
Try Wolfram alpha
– Lorenzo
Jul 18 at 13:59
@Bruce, I didn't find any multiple integral$n>2$ in this book. It is a very good book, but I want a cookbook just for reference.
– user85361
Jul 18 at 14:21
1
This is the closet I could get it wolframalpha.com/input/… more woflram-fu might be able to get you further. E.g. you might want to convert it to polar coordinates first, and have wolfram evaluate the 1 variable integral.
– Lorenzo
Jul 18 at 15:18
1
@user85361 Your integral is radially symmetric, so you can compute the integral in the radial direction separately from the rest. E.g. as in here: en.wikipedia.org/wiki/…
– Lorenzo
Jul 19 at 5:47
 |Â
show 5 more comments
Try Paul J Nahin's book ... Inside Interesting Integrals ... to learn amazing techniques to solve these sorts of integrals
– Bruce
Jul 18 at 13:58
1
Try Wolfram alpha
– Lorenzo
Jul 18 at 13:59
@Bruce, I didn't find any multiple integral$n>2$ in this book. It is a very good book, but I want a cookbook just for reference.
– user85361
Jul 18 at 14:21
1
This is the closet I could get it wolframalpha.com/input/… more woflram-fu might be able to get you further. E.g. you might want to convert it to polar coordinates first, and have wolfram evaluate the 1 variable integral.
– Lorenzo
Jul 18 at 15:18
1
@user85361 Your integral is radially symmetric, so you can compute the integral in the radial direction separately from the rest. E.g. as in here: en.wikipedia.org/wiki/…
– Lorenzo
Jul 19 at 5:47
Try Paul J Nahin's book ... Inside Interesting Integrals ... to learn amazing techniques to solve these sorts of integrals
– Bruce
Jul 18 at 13:58
Try Paul J Nahin's book ... Inside Interesting Integrals ... to learn amazing techniques to solve these sorts of integrals
– Bruce
Jul 18 at 13:58
1
1
Try Wolfram alpha
– Lorenzo
Jul 18 at 13:59
Try Wolfram alpha
– Lorenzo
Jul 18 at 13:59
@Bruce, I didn't find any multiple integral$n>2$ in this book. It is a very good book, but I want a cookbook just for reference.
– user85361
Jul 18 at 14:21
@Bruce, I didn't find any multiple integral$n>2$ in this book. It is a very good book, but I want a cookbook just for reference.
– user85361
Jul 18 at 14:21
1
1
This is the closet I could get it wolframalpha.com/input/… more woflram-fu might be able to get you further. E.g. you might want to convert it to polar coordinates first, and have wolfram evaluate the 1 variable integral.
– Lorenzo
Jul 18 at 15:18
This is the closet I could get it wolframalpha.com/input/… more woflram-fu might be able to get you further. E.g. you might want to convert it to polar coordinates first, and have wolfram evaluate the 1 variable integral.
– Lorenzo
Jul 18 at 15:18
1
1
@user85361 Your integral is radially symmetric, so you can compute the integral in the radial direction separately from the rest. E.g. as in here: en.wikipedia.org/wiki/…
– Lorenzo
Jul 19 at 5:47
@user85361 Your integral is radially symmetric, so you can compute the integral in the radial direction separately from the rest. E.g. as in here: en.wikipedia.org/wiki/…
– Lorenzo
Jul 19 at 5:47
 |Â
show 5 more comments
1 Answer
1
active
oldest
votes
up vote
1
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accepted
In polar coordinates, we have the following formula for the volume form:
$dV = r^n-1 sin^n-2 ( phi_1) ldots sin(phi_n-2) dr dphi_1 ldots dphi_n-1 = r^n-1 S dr dPhi$, where $dPhi = dphi_1 ldots dphi_n-1$ and $S$ is the function of $phi_1, ldots, phi_n-2$ appearing in the integrand.
See: https://en.wikipedia.org/wiki/N-sphere#Spherical_coordinates
Let $f(x) = ||x||^3 exp( - a||x||^2) = r^3 exp( - ar^2)$
Hence $int_R^n f(x) dx = int_S^n-1 ( int_0^infty f(r) r^n-1 dr ) S d Phi = ( int_0^infty f(r) r^n-1 dr ) (int_S^n-1 S dphi) = ( int_0^infty f(r) r^n-1 dr ) (textsurface area of n-1 sphere)$.
Wolfram tells us $int_0^infty f(r) r^n-1 dr = int_0^infty r^n + 2 exp(- ar^2) dr = frac12 a^-n/2 - 3/2 Gamma((n + 3)/2)$: http://www.wolframalpha.com/input/?i=integrate+r%5E3+e%5E%7B-ar%5E2%7D+r%5E%7Bn-1%7D+from+0+to+infinity
(You could probably do this integral by hand.)
Wolfram also knows the surface area of the sphere: http://www.wolframalpha.com/input/?i=surface+area+of+n+dimensional+sphere
Multiplying the two gives us the formula in your question: here
Let me know if any steps are unclear. :)
answered Jul 21 at 21:26
Lorenzo
11.5k31537
Thank you very much. I know that $dv$ formula comes from determinant of Jacobian of the transformation of variables, but I think it will be much clearer for later reading of your good answer to add it to the question.
– user85361
Jul 22 at 5:49
Would you please take a look at this question: math.stackexchange.com/questions/2858454/… . Thank you
– user85361
Jul 22 at 5:51
add a comment |Â
1 Answer
1
active
oldest
votes
1 Answer
1
active
oldest
votes
active
oldest
votes
active
oldest
votes
up vote
1
down vote
accepted
In polar coordinates, we have the following formula for the volume form:
$dV = r^n-1 sin^n-2 ( phi_1) ldots sin(phi_n-2) dr dphi_1 ldots dphi_n-1 = r^n-1 S dr dPhi$, where $dPhi = dphi_1 ldots dphi_n-1$ and $S$ is the function of $phi_1, ldots, phi_n-2$ appearing in the integrand.
See: https://en.wikipedia.org/wiki/N-sphere#Spherical_coordinates
Let $f(x) = ||x||^3 exp( - a||x||^2) = r^3 exp( - ar^2)$
Hence $int_R^n f(x) dx = int_S^n-1 ( int_0^infty f(r) r^n-1 dr ) S d Phi = ( int_0^infty f(r) r^n-1 dr ) (int_S^n-1 S dphi) = ( int_0^infty f(r) r^n-1 dr ) (textsurface area of n-1 sphere)$.
Wolfram tells us $int_0^infty f(r) r^n-1 dr = int_0^infty r^n + 2 exp(- ar^2) dr = frac12 a^-n/2 - 3/2 Gamma((n + 3)/2)$: http://www.wolframalpha.com/input/?i=integrate+r%5E3+e%5E%7B-ar%5E2%7D+r%5E%7Bn-1%7D+from+0+to+infinity
(You could probably do this integral by hand.)
Wolfram also knows the surface area of the sphere: http://www.wolframalpha.com/input/?i=surface+area+of+n+dimensional+sphere
Multiplying the two gives us the formula in your question: here
Let me know if any steps are unclear. :)
answered Jul 21 at 21:26
Lorenzo
11.5k31537
Thank you very much. I know that $dv$ formula comes from determinant of Jacobian of the transformation of variables, but I think it will be much clearer for later reading of your good answer to add it to the question.
– user85361
Jul 22 at 5:49
Would you please take a look at this question: math.stackexchange.com/questions/2858454/… . Thank you
– user85361
Jul 22 at 5:51
add a comment |Â
up vote
1
down vote
accepted
In polar coordinates, we have the following formula for the volume form:
$dV = r^n-1 sin^n-2 ( phi_1) ldots sin(phi_n-2) dr dphi_1 ldots dphi_n-1 = r^n-1 S dr dPhi$, where $dPhi = dphi_1 ldots dphi_n-1$ and $S$ is the function of $phi_1, ldots, phi_n-2$ appearing in the integrand.
See: https://en.wikipedia.org/wiki/N-sphere#Spherical_coordinates
Let $f(x) = ||x||^3 exp( - a||x||^2) = r^3 exp( - ar^2)$
Hence $int_R^n f(x) dx = int_S^n-1 ( int_0^infty f(r) r^n-1 dr ) S d Phi = ( int_0^infty f(r) r^n-1 dr ) (int_S^n-1 S dphi) = ( int_0^infty f(r) r^n-1 dr ) (textsurface area of n-1 sphere)$.
Wolfram tells us $int_0^infty f(r) r^n-1 dr = int_0^infty r^n + 2 exp(- ar^2) dr = frac12 a^-n/2 - 3/2 Gamma((n + 3)/2)$: http://www.wolframalpha.com/input/?i=integrate+r%5E3+e%5E%7B-ar%5E2%7D+r%5E%7Bn-1%7D+from+0+to+infinity
(You could probably do this integral by hand.)
Wolfram also knows the surface area of the sphere: http://www.wolframalpha.com/input/?i=surface+area+of+n+dimensional+sphere
Multiplying the two gives us the formula in your question: here
Let me know if any steps are unclear. :)
answered Jul 21 at 21:26
Lorenzo
11.5k31537
Thank you very much. I know that $dv$ formula comes from determinant of Jacobian of the transformation of variables, but I think it will be much clearer for later reading of your good answer to add it to the question.
– user85361
Jul 22 at 5:49
Would you please take a look at this question: math.stackexchange.com/questions/2858454/… . Thank you
– user85361
Jul 22 at 5:51
add a comment |Â
up vote
1
down vote
accepted
up vote
1
down vote
accepted
In polar coordinates, we have the following formula for the volume form:
$dV = r^n-1 sin^n-2 ( phi_1) ldots sin(phi_n-2) dr dphi_1 ldots dphi_n-1 = r^n-1 S dr dPhi$, where $dPhi = dphi_1 ldots dphi_n-1$ and $S$ is the function of $phi_1, ldots, phi_n-2$ appearing in the integrand.
See: https://en.wikipedia.org/wiki/N-sphere#Spherical_coordinates
Let $f(x) = ||x||^3 exp( - a||x||^2) = r^3 exp( - ar^2)$
Hence $int_R^n f(x) dx = int_S^n-1 ( int_0^infty f(r) r^n-1 dr ) S d Phi = ( int_0^infty f(r) r^n-1 dr ) (int_S^n-1 S dphi) = ( int_0^infty f(r) r^n-1 dr ) (textsurface area of n-1 sphere)$.
Wolfram tells us $int_0^infty f(r) r^n-1 dr = int_0^infty r^n + 2 exp(- ar^2) dr = frac12 a^-n/2 - 3/2 Gamma((n + 3)/2)$: http://www.wolframalpha.com/input/?i=integrate+r%5E3+e%5E%7B-ar%5E2%7D+r%5E%7Bn-1%7D+from+0+to+infinity
(You could probably do this integral by hand.)
Wolfram also knows the surface area of the sphere: http://www.wolframalpha.com/input/?i=surface+area+of+n+dimensional+sphere
Multiplying the two gives us the formula in your question: here
Let me know if any steps are unclear. :)
answered Jul 21 at 21:26
Lorenzo
11.5k31537
In polar coordinates, we have the following formula for the volume form:
$dV = r^n-1 sin^n-2 ( phi_1) ldots sin(phi_n-2) dr dphi_1 ldots dphi_n-1 = r^n-1 S dr dPhi$, where $dPhi = dphi_1 ldots dphi_n-1$ and $S$ is the function of $phi_1, ldots, phi_n-2$ appearing in the integrand.
See: https://en.wikipedia.org/wiki/N-sphere#Spherical_coordinates
Let $f(x) = ||x||^3 exp( - a||x||^2) = r^3 exp( - ar^2)$
Hence $int_R^n f(x) dx = int_S^n-1 ( int_0^infty f(r) r^n-1 dr ) S d Phi = ( int_0^infty f(r) r^n-1 dr ) (int_S^n-1 S dphi) = ( int_0^infty f(r) r^n-1 dr ) (textsurface area of n-1 sphere)$.
Wolfram tells us $int_0^infty f(r) r^n-1 dr = int_0^infty r^n + 2 exp(- ar^2) dr = frac12 a^-n/2 - 3/2 Gamma((n + 3)/2)$: http://www.wolframalpha.com/input/?i=integrate+r%5E3+e%5E%7B-ar%5E2%7D+r%5E%7Bn-1%7D+from+0+to+infinity
(You could probably do this integral by hand.)
Wolfram also knows the surface area of the sphere: http://www.wolframalpha.com/input/?i=surface+area+of+n+dimensional+sphere
Multiplying the two gives us the formula in your question: here
Let me know if any steps are unclear. :)
answered Jul 21 at 21:26
Lorenzo
11.5k31537
answered Jul 21 at 21:26
Lorenzo
11.5k31537
answered Jul 21 at 21:26
Lorenzo
11.5k31537
answered Jul 21 at 21:26
Lorenzo
11.5k31537
11.5k31537
Thank you very much. I know that $dv$ formula comes from determinant of Jacobian of the transformation of variables, but I think it will be much clearer for later reading of your good answer to add it to the question.
– user85361
Jul 22 at 5:49
Would you please take a look at this question: math.stackexchange.com/questions/2858454/… . Thank you
– user85361
Jul 22 at 5:51
add a comment |Â
Thank you very much. I know that $dv$ formula comes from determinant of Jacobian of the transformation of variables, but I think it will be much clearer for later reading of your good answer to add it to the question.
– user85361
Jul 22 at 5:49
Would you please take a look at this question: math.stackexchange.com/questions/2858454/… . Thank you
– user85361
Jul 22 at 5:51
Thank you very much. I know that $dv$ formula comes from determinant of Jacobian of the transformation of variables, but I think it will be much clearer for later reading of your good answer to add it to the question.
– user85361
Jul 22 at 5:49
Thank you very much. I know that $dv$ formula comes from determinant of Jacobian of the transformation of variables, but I think it will be much clearer for later reading of your good answer to add it to the question.
– user85361
Jul 22 at 5:49
Would you please take a look at this question: math.stackexchange.com/questions/2858454/… . Thank you
– user85361
Jul 22 at 5:51
Would you please take a look at this question: math.stackexchange.com/questions/2858454/… . Thank you
– user85361
Jul 22 at 5:51
add a comment |Â
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Try Paul J Nahin's book ... Inside Interesting Integrals ... to learn amazing techniques to solve these sorts of integrals
– Bruce
Jul 18 at 13:58
1
Try Wolfram alpha
– Lorenzo
Jul 18 at 13:59
@Bruce, I didn't find any multiple integral$n>2$ in this book. It is a very good book, but I want a cookbook just for reference.
– user85361
Jul 18 at 14:21
1
This is the closet I could get it wolframalpha.com/input/… more woflram-fu might be able to get you further. E.g. you might want to convert it to polar coordinates first, and have wolfram evaluate the 1 variable integral.
– Lorenzo
Jul 18 at 15:18
1
@user85361 Your integral is radially symmetric, so you can compute the integral in the radial direction separately from the rest. E.g. as in here: en.wikipedia.org/wiki/…
– Lorenzo
Jul 19 at 5:47