Simplifying rotational invariant integrals with spherical coordinates

[mhill]

let be the integral

$$\int_{R^{3}}d^{3}r F( \vec r . \vec r , \vec r . \vec a , |r| ,|a|)$$ (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 $$\int d\Omega \int_{0}^{\infty}k^{2}F(k)dk$$ being k=|r| (modulus of r)

i believe answer is affirmative

 

[mathman]

let be the integral

$$\int_{R^{3}}d^{3}r F( \vec r . \vec r , \vec r . \vec a , |r| ,|a|)$$ (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 $$\int d\Omega \int_{0}^{\infty}k^{2}F(k)dk$$ 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.