Vector & Tensor Transformation in Physics

[nigelscott]

I'm not sure if this belongs in physics or here. Consider the transformation of a vector from one coordinate system to the other. I can write:

Vⁿ = (∂yⁿ/∂x^m)V^m - contravariant form

V_n = (∂x^m/∂yⁿ)V_m - covariant form

In each case are the partials equivalent to the Jacobean matrices? Also, what about the case of a tensor

T_mn = (∂x^r/∂y^m)(∂x^s/∂yⁿ)T_rs

Is the transformation just the product of 2 Jacobeans?

 

[Bacle2]

Yes, the Jacobian is used to describe coordinate changes. Is that what you

were asking?

 

[nigelscott]

Yes. I wasn't quite sure if there were any implications associated with covariant versus contravariant vectors in differential geometry. Also, in the case of a tensor, I wasn't sure if the transformation was the product of the Jacobean with it's inverse or transpose.