Vectors and covectors under change of coordinates

[spaghetti3451]

Homework Statement

If $\bf{v}$ is a vector and $\alpha$ is a covector, compute directly in coordinates that $\sum a_{i}^{V}v^{i}_{V}=\sum a_{i}^{U}v^{j}_{U}$.

What happens if $\bf{w}$ is another vector and one considers $\sum v^{i}w^{i}$?

Homework Equations

The Attempt at a Solution

$\alpha(\bf{v})=\alpha(\bf{v})$

$\implies (a_{i}^{V}\ \sigma^{i}_{V})(v^{k}_{V}\ \vec{e}_{k}^{V})=(a_{j}^{U}\ \sigma^{j}_{U})( v^{l}_{U}\ \vec{e}_{l}^{U})$,

where the Einstein summation convention has used to sum over the $i$, $j$, $k$ and $l$ indices and the left-hand side uses the $U$ coordinate system and the right-hand side uses the $V$ coordinate system, so that

$\implies a_{i}^{V}\ v^{k}_{V}\ \sigma^{i}_{V}(\vec{e}_{k}^{V}) = a_{j}^{U}\ v^{l}_{U}\ \sigma^{j}_{U}(\vec{e}_{l}^{U})$

$\implies a_{i}^{V}\ v^{k}_{V}\ \delta^{i}_{k} = a_{j}^{U}\ v^{l}_{U}\ \delta^{j}_{l}$

$\implies a_{i}^{V}\ v^{i}_{V}\ = a_{j}^{U}\ v^{j}_{U}$For the second part, again using the Einstein summation convention,

$v^{i}w^{i}=v^{i}w_{j}\delta^{ji}=\langle \vec{v},\vec{w}\rangle$ for $g^{ij}=\delta^{ij}$.

Now, $g^{ij}=\delta^{ij}$ is true only for Cartesian coordinates in flat space, so that the metric changes under a coordinate transformation. Therefore, the transformation of $v^{i}w^{i}$ depends upon the specific change of coordinates.

What do you think?

 

[Lucas SV]

The first part is good. In the second part, I think they want you to show that $v^i w^i$ is invariant under a change of basis. Also if you want to talk about curvilinear coordinates (more generally curved spaces a.k.a. manifolds), you should distinguish vectors from points, and then one usually considers vector fields. There is a reason why a linear algebra course will not have a chapter on curvilinear coordinates. Yet to study curvilinear coordinates (and more generally, manifolds), you need to know about vector spaces. Take as an example, $\mathbb{R}^2$. If this is thought as a two dimensional vector space, we don't talk about coordinate systems (since we don't care about general coordinate transformations), instead we talk about basis. But when you start to think of the Euclidean plane as a two dimensional manifold (albeit a very simple one), then you talk about general coordinate systems.You probably should state in the problem statement what the $U$ and $V$ subscripts/superscripts actually mean. It is just nicer for the reader if you make it a habit to state what your symbols mean, right after you write them down. An exception to this is if there is a conventional notation that you expect the reader to know. However since, the notation you used is not a widely used convention, you should state it.