ehrenfest said:Homework Statement
I am trying to show that
[tex] \vec{e'}_a = \frac{\partial x^b}{\partial x'^a} \vec{e}_b [/tex]
where the e's are bases on a manifold and the primes mean a change of coordinates
Yes, I am dealing with coordinate bases.George Jones said:You're dealing with coodinates bases, or else your expression above isn't true. Thus
[tex]\vec{e}_b = \frac{\partial}{\partial x^b}[/tex]
[tex]\vec{e'}_a = \frac{\partial}{\partial x'^a}.[/tex]
Basis vectors on manifolds are a set of vectors that define the tangent space at a particular point on a manifold. They are used to represent the local coordinate system of a point on a manifold.
Transformations of basis vectors are important because they allow us to understand how the tangent space changes as we move from one point to another on a manifold. This is crucial for understanding the geometry and topology of the manifold.
A coordinate basis is a set of vectors that are aligned with the coordinate axes of the local coordinate system at a point on the manifold. A frame basis, on the other hand, is a set of vectors that span the tangent space at a point on the manifold, but are not necessarily aligned with the coordinate axes.
The transformation of basis vectors on a manifold is calculated using the Jacobian matrix, which represents the linear transformation between coordinate systems. The Jacobian matrix is calculated by taking the derivatives of the coordinate system transformation equations.
The metric tensor is a mathematical object that defines the inner product between vectors in the tangent space of a manifold. It is used to measure angles, lengths, and volumes on the manifold, and is crucial for understanding the geometry and curvature of the manifold.