- #1

harmyder

- 33

- 1

## Homework Statement

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.##

## Homework Equations

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

## 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.

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