Triple integral

  • Thread starter mhill
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  • #1
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let be the integral

[tex] \int_{R^{3}}d^{3}r F( \vec r . \vec r , \vec r . \vec a , |r| ,|a|) [/tex] (1)

F depends only on the scalar product of vector r=(x,y,z) and its modulus |r| , hence it is invariant under rotation and traslations (since scalar product is invariant under rotation and traslation) my question is if using polar spherical coordinates we can put the integral (1)

as [tex] \int d\Omega \int_{0}^{\infty}k^{2}F(k)dk [/tex] being k=|r| (modulus of r)

i believe answer is affirmative
 

Answers and Replies

  • #2
mathman
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let be the integral

[tex] \int_{R^{3}}d^{3}r F( \vec r . \vec r , \vec r . \vec a , |r| ,|a|) [/tex] (1)

F depends only on the scalar product of vector r=(x,y,z) and its modulus |r| , hence it is invariant under rotation and traslations (since scalar product is invariant under rotation and traslation) my question is if using polar spherical coordinates we can put the integral (1)

as [tex] \int d\Omega \int_{0}^{\infty}k^{2}F(k)dk [/tex] being k=|r| (modulus of r)

i believe answer is affirmative
Your assumption is correct. I suspect your original statement refers to a scalar product of r and a. Also I assume a is a constant vector.
 

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