Covariant and Contravariant Vector

AA23

New member
I have been given the following problem:

The covariant vector field is:

$$v_{i}$$ = \begin{matrix} x+y\\ x-y\end{matrix}

What are the components for this vector field at (4,1)?

$$v_{i}$$ = \begin{matrix} 5\\ 3\end{matrix}

Now I can use this information to solve the following:

$$\bar{V_\alpha}$$

But am unsure for $$\bar{V^\alpha}$$.

I imagine it would be a similar approach with a few changes. Any help would be brilliant thank you Fantini

MHB Math Helper
If you mean to write a matrix, you could have used
Code:
\begin{bmatrix} 1 & 2 \\ 3 & 4 \end{bmatrix}
to output
$$\begin{bmatrix} 1 & 2 \\ 3 & 4 \end{bmatrix}$$

Let us fix some notation here: are you using lower indexes to indicate covariant tensors and upper to indicate contravariant, or the other way around? Also, what are $\overline{V_\alpha}$ and $\overline{V}^{\alpha}$? I'm a little confused, if you could clarify perhaps we could arrive at an answer together. At a first glance though, your calculations look correct, although I can't see where they headed because I don't understand what the symbols are meant to represent.

AA23

New member
My calculations are quite long and would take me a very long time to write them out in LaTeX. I would like to include an attachment but am unsure how to delete old ones to make room.

Thanks Fantini

MHB Math Helper
Hey AA23, you still haven't answered my last questions: what do these $\overline{V_\alpha}$, $\overline{V}^{\alpha}$ mean? Also, it seemed you had

$$v_i = \begin{bmatrix} x+y \\ x-y \end{bmatrix},$$

to which you applied at the point $(4,1)$, getting

$$v_i (4,1) = \begin{bmatrix} 5+1 \\ 5-1 \end{bmatrix}.$$

Are there other calculations? Also, we let column matrices denote vectors and we use line matrices to denote covetors, so perhaps that would be

$$v^i = \begin{bmatrix} x+y & x-y \end{bmatrix} .$$

AA23

New member
$$\bar{V_{\alpha}}$$ represents a covariant tensor

$$\bar{V^{\alpha}}$$ represents a contravariant tensor

Yes I can solve the question to find the components of

$$\bar{V_{\alpha}}$$

But come stuck when finding them for

$$\bar{V^{\alpha}}$$