How are Vectors described in Bispherical Coordinates?

[tade]

I was reading a paper that described a vector field in terms of its three components , $A_σ,A_τ,A_φ$.
with σ, τ and φ being the three bispherical coordinates.

what does $A_σ$ mean? In what direction does the component point? Likewise for the other two components.

 

[Simon Bridge]

Have you tried looking up the definition?

 

[tade]

Have you tried looking up the definition?

yes but to no avail. I couldn't find anything remotely related.

in bipolar coordinates, is the σ component the component of the vector tangential to the σ circle?

I can't be certain about how it works in 3D bispherical coordinates though.

also, in bipolar coordinates the τ and σ circles intersect at two points, how does one specify a specific point?

 

[chiro]

Hey tade.

Usually it means a vector basis component if it is written in that form.

In differential geometry, the basis vectors are curved instead of straight and follow what is called a geodesic or a curved path instead of a straight line.

 

[tade]

Hey tade.

Usually it means a vector basis component if it is written in that form.

In differential geometry, the basis vectors are curved instead of straight and follow what is called a geodesic or a curved path instead of a straight line.

I see. that sounds rather confusing. do you have a diagram that shows how to obtain $A_σ,A_τ,A_φ$ from a "straight arrow" Euclidean vector?

The vector field in question is an electric field. Standard electrostatics. Though the paper uses bispherical coordinates due to the nature of the system.

 

[chiro]

In normal geometry the arrows are straight and you write a point as a linear combination of them.

In differential geometry the arrows are curved and instead of doing distance via the normal metric you use a thing like a metric tensor to find the distance between two points.

Basically the vectors are curved which means you have to invoke differential geometry and look at the distance in terms of arc-length across a manifold instead of the standard distance formula via the Pythagorean theorem and inner products in R^n.

 

[tade]

Also, in bipolar coordinates the τ and σ circles intersect at two points, how does one specify a specific point?

In normal geometry the arrows are straight and you write a point as a linear combination of them.

In differential geometry the arrows are curved and instead of doing distance via the normal metric you use a thing like a metric tensor to find the distance between two points.

Basically the vectors are curved which means you have to invoke differential geometry and look at the distance in terms of arc-length across a manifold instead of the standard distance formula via the Pythagorean theorem and inner products in R^n.

I'm still confused, so I want to cut to the chase. Am I allowed to post a scientific paper here? Its only 3 pages. It'll help to get my point across.

 

[chiro]

You can definitely try and post a link.

If the moderators will ban the link then they will do so but I don't see the harm in showing us the information.

 

[tade]

 

[tade]

Imagine a line that connects the centers of both spheres. I want to know what the value of the electric field along this line is when its magnitude is at its strongest.

I believe that the field is at its strongest at the surface of either sphere . The equations for the electric field have been derived but idk how to make sense of them.

 

[tade]

You can definitely try and post a link.

If the moderators will ban the link then they will do so but I don't see the harm in showing us the information.

just a heads up that I've replied.