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Intrinsic derivative of constant vector field along a curve

  1. Jan 24, 2017 #1
    1. The problem statement, all variables and given/known data
    Suppose that ##T_i## is the contravariant component of a vector field ##\mathbf{T}## that is constant along the trajectory ##\gamma.## Show that intrinsic derivative is ##0.##

    2. Relevant equations

    $$\frac{\delta T_i}{\delta t} = \frac{dT^i}{dt}+V^j\Gamma^i_{jk}T^k$$

    3. The attempt at a solution

    $$\begin{align}\mathbf{T} = T^i \mathbb{Z}_i\\T^i = \frac{d\mathbf{T}}{dZ_i}\label{ti}\end{align}$$
    But from ##\ref{ti}## i see that ##T_i=0.## Probably, ##\ref{ti}## is wrong.

    Another attempt:)
    $$\begin{align}
    \mathbf{T} &= T^i \mathbb{Z}_i\\
    \mathbf{T}\cdot\mathbb{Z}^i &= T^i\\
    \frac{dT^i}{dt}&= \frac{d\mathbf{T}}{dt}\mathbb{Z}^i + \mathbf{T}\frac{\partial\mathbb{Z}^i}{\partial Z^j}\frac{dZ^j}{dt}\\
    &=-\mathbf{T}\Gamma^i_{jk}\mathbb{Z}^k\frac{dZ^j}{dt}\\
    &=-\mathbf{T}\mathbb{Z}^k\Gamma^i_{jk} V^j\\
    &=-T^k\Gamma^i_{jk}V^j
    \end{align}$$

    OMH, looks like i have solved it while writing it here. Just need a confirmation.
     
    Last edited: Jan 24, 2017
  2. jcsd
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