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## Homework Statement

A tensor and vector have components T

^{α}

_{β}

^{γ}, and v

^{α}respectively in a coordinate system x

^{μ}. There is another coordinate system x'

^{μ}. Show that T

^{α}

_{β}

^{γ}v

^{β}= T

^{α}

^{β}

^{γ}v

_{β}

## Homework Equations

umm not sure...

∇

_{α}v

_{β}= ∂v

^{β}/∂x

^{α}- Γ

^{γ}

_{α}

_{β}v

_{γ}

## The Attempt at a Solution

T

^{α}

_{β}

^{γ}v

^{β}= (∂x

^{α}/∂x

^{i}*∂x

^{j}/∂x

^{β}*∂x

^{γ}/∂x

^{k}*T

^{i}

_{j}

^{k})(∂x

^{β}/∂x

^{a}*v

^{a})

T

^{α}

^{β}

^{γ}v

_{β}= (∂x

^{α}/∂x

^{i}*∂x

^{β}/∂x

^{j}*∂x

^{γ}/∂x

^{k}*T

^{i}

^{j}

^{k})(∂x

^{a}/∂x

^{β}*v

^{a})

and the two are not equal which they should be. I really don't know where I've went wrong...