- #1

tetris11

- 23

- 0

## Homework Statement

I know this is an easy question, I just can't seem to grasp what I am actually doing:

Let M be a manifold.

Let V

^{a}be contravariant, and W

_{a}be covariant.

Show that [tex]\mu[/tex]=V

^{a}W

_{a}

## Homework Equations

(couldn't get Latex to work consistently, sorry)

(1) V

^{ 'a}= (dx

^{ 'a}/ dx

^{b}) V

^{b}

(2) W

^{ '}

_{a}= (dx

^{b}/ dx

^{ 'a}) W

_{b}

(3) A

^{c}D

_{c}= Sum of A

^{c}D

_{c}from c=1...n

## The Attempt at a Solution

Well I just multiplied (1) and (2) together to get:

V

^{ 'a}W

^{ '}

_{a}= (dx

^{ 'a}/ dx

^{b}) (dx

^{b}/ dx

^{ 'a}) V

^{b}W

_{b}

The brackets cancel out and I get:

V

^{ 'a}W

^{ '}

_{a}=V

^{b}W

_{b}= Sum of V

^{b}W

_{b}from b=1...n (from (3))

How this proves anything is scalar is beyond me.

Please help!