- #1

guv

- 123

- 22

## Homework Statement

Hi I am reviewing the following document on tensor:

https://www.grc.nasa.gov/www/k-12/Numbers/Math/documents/Tensors_TM2002211716.pdf

## Homework Equations

In the middle of page 27, the author says:

Now, using the covariant representation, the expression $$\vec V=\vec V^*$$

then becomes

$$\vec V = V_j \vec e^{(j)}= V_j^* \vec e^{(j)*} = \vec V^*$$

## The Attempt at a Solution

How does this work? Earlier in the document, the Vector is always represented as

$$\vec V = V_j \vec e_{(j)} = V^j \vec e^{(j)} $$

I don't see how suddenly the coordinates or components of covariant basis combing with the contracovariant basis can represent the same vector? It's been made clear earlier,

$$V_j = V^i g_{ij}$$ and $$g_{ij}$$ is not 1 in general.

Thank you for clarification.