Dot product for vectors in spherical coordinates

[Wminus]

Hi all.

I'm struggling with taking dot products between vectors in spherical coordinates. I just cannot figure out how to take the dot product between two arbitrary spherical-coordinate vectors $\bf{v_1}$ centered in $(r_1,\theta_1,\phi_1)$ and $\bf{v_2}$ centered in $(r_2,\theta_2,\phi_2)$ without converting them to cartesian coordinates first.

Could you guys please help me? The main issue is that the basis for $v_1$ and $v_2$ are different so everything becomes super complicated.

Thanks

 

[Wminus]

By the way, if this requires advanced mathematics and is very complicated, please just tell me that's the case instead of spending your time writing a long post. I don't really have the time to dwell too long on this issue.

 

[mathman]

Your last remark is confusing. What do you mean by "basis for v₁ and v₂ are different"?

 

[SteamKing]

Hi all.

I'm struggling with taking dot products between vectors in spherical coordinates. I just cannot figure out how to take the dot product between two arbitrary spherical-coordinate vectors $\bf{v_1}$ centered in $(r_1,\theta_1,\phi_1)$ and $\bf{v_2}$ centered in $(r_2,\theta_2,\phi_2)$ without converting them to cartesian coordinates first.

Could you guys please help me? The main issue is that the basis for $v_1$ and $v_2$ are different so everything becomes super complicated.

Thanks

I don't think there is a definition of the dot (scalar) product which involves using vectors expressed in spherical coordinates, at least, not one which makes sense.

What started you down this dark road in the first place?

 

[Wminus]

Your last remark is confusing. What do you mean by "basis for v₁ and v₂ are different"?

The basis vectors $\hat{\phi}$, $\hat{\theta}$ depend on the angles $\phi$,$\theta$. This is what I meant with the basis are different for vectors centered in different points of spherical space. Sorry for the confusion.

I don't think there is a definition of the dot (scalar) product which involves using vectors expressed in spherical coordinates, at least, not one which makes sense.

What started you down this dark road in the first place?

I was derailed while doing an assignment :( . I ended up wanting to calculate $\vec{L} \cdot \vec{L}$ where $\vec{L} = -i \hbar \vec{r} \times \nabla$ is the angular momentum operator in spherical coordinates.

 

[PWiz]

The basis vectors ϕ^\hat{\phi}, θ^\hat{\theta} depend on the angles ϕ\phi,θ\theta. This is what I meant with the basis are different for vectors centered in different points of spherical space. Sorry for the confusion.

The main problem is that in spherical coordinates, the local orthonormal basis are not the global coordinate basis, and hence you cannot obtain a 'neat' expression for the dot product using them. You can obtain an expression in terms of them using Cartesian conversions, but the expression is long, and it would be better to simply change coordinates first and then perform the dot product.