Sherical distribution as a tensor?

[Dr Bwts]

Hi,

I have a 3D spherical distribution of mass in polar co-ordinates. The directional cosines of any point in the distribution are l, m and n. From which a 3x3 symetrical matrix, T, is constructed (shown below) using the idea of moments of inertia about a vector, derivation not shown here.

l = sin θ cos ψ
m = sin θ sin ψ
n = cos θ

T = [ Ʃl^2 Ʃlm Ʃln
Ʃlm Ʃm^2 Ʃmn
Ʃln Ʃmn Ʃn^2 ]

My question, is T a tensor? If so how would I proove this? Invariants?

Thanks

Nic

 

[Valerie Park]

Yes, T is a tensor. To prove this, you can consider the transformation of T under a rotation in space. Since T is a 3x3 symmetric matrix, it will transform according to an invariant tensor representation. This means that the components of T will remain unchanged when the coordinates of the points in the distribution are rotated. This invariance under rotation is the defining property of a tensor.