I Indices Up & Down: Understanding Dual Vectors

[LagrangeEuler]

I found notation in the form
$\vec{e}_{\alpha} \cdot \vec{x}=x^{\alpha}$
and also
$\langle \vec{e}^{* \alpha}, \vec{x} \rangle =x^{\alpha}$.
If I understand this $\vec{e}_{\alpha}$ is unit vector in direct space, and $\vec{e}^{* \alpha}$ is unit vector in dual space, and the bracket $\langle, \rangle$ is like dual vector on vector on direct space. But why result is $x^{\alpha}$?

 

[martinbn]

The $x^\alpha$ is the $\alpha$-component of $\vec{x}$ in the basis $\{\vec{e}_\alpha\}$. So $\vec{x}=x^\alpha\vec{e}_\alpha$, the rest is the definition of the inner product and what a dual basis is.

 

[Orodruin]

I found notation in the form
$\vec{e}_{\alpha} \cdot \vec{x}=x^{\alpha}$

This is true only in an orthonormal basis. In general,
$$
\vec e_\alpha \cdot \vec x = \vec e_\alpha \cdot x^\beta \vec e_\beta = g_{\alpha\beta} x^\beta.
$$

 

[LagrangeEuler]

Ok. Tnx. I am liitle bit confused with notation.
\frac{\partial}{\partial x^{\mu}}=\partial x_{\mu}?
So if I write
\frac{\partial}{\partial x^{\mu}} x^{\nu}=\partial x_{\mu}x^{\nu}=\delta_{\mu}^{\nu}.
So if I have for example
\frac{\partial}{\partial x_{\mu}}=\partial x^{\mu}?
Is it correct?

 

[Orodruin]

Ok. Tnx. I am liitle bit confused with notation.
\frac{\partial}{\partial x^{\mu}}=\partial x_{\mu}?

This is not standard notation. The standard notation is
$$
\frac{\partial}{\partial x^\mu} = \partial_\mu.
$$

So if I write
\frac{\partial}{\partial x^{\mu}} x^{\nu}=\partial x_{\mu}x^{\nu}=\delta_{\mu}^{\nu}.

It is not so much a question about writing as it is about performing the derivative, but yes, $\partial_\mu x^\nu = \delta^\nu_\mu$.

So if I have for example
\frac{\partial}{\partial x_{\mu}}=\partial x^{\mu}?
Is it correct?

This would be $\partial/\partial x_\mu = \partial^\mu$.

 

[Sorcerer]

Side-note: shoudn't this be I level and not B?

 

[Sorcerer]

This is not standard notation. The standard notation is
$$
\frac{\partial}{\partial x^\mu} = \partial_\mu.
$$

Question: is that because

$$
\frac{\partial}{\partial x^\mu}$$

is an operator, while

$$\partial x_\mu.$$

would be a specific operation on x_μ?

Trying to learn these terms. Thanks.

 

[Nugatory]

Side-note: shoudn't this be I level and not B?

Yes, good point. Fixed. (And you can report these instead of asking in thread - generally gets a faster response)

 

[Orodruin]

while

$$\partial x_\mu.$$

would be a specific operation on x_μ?

This notation does not make sense to me at all. You would first have to identify what you mean by just $\partial$ without the index.

 

[Sorcerer]

This notation does not make sense to me at all. You would first have to identify what you mean by just $\partial$ without the index.

Well, I was seeing it as just a differential of x_μ. Is there such thing as just a partial differential, in the sense of say, dx, dy, dz?

But my other question, is that right? It's an operator?

 

[Orodruin]

That operator is completely different and typically denoted with d, not $\partial$. The one-forms $dx^\mu$ form a basis of the dual space.

 

[PeroK]

Question: is that because

$$
\frac{\partial}{\partial x^\mu}$$

is an operator, while

$$\partial x_\mu.$$

would be a specific operation on x_μ?

Trying to learn these terms. Thanks.

To some extent you have to learn the specific notation and accept it. You can't necessarily deduce what a shorthand means from some absolute first principles.

The standard notation is
$$
\frac{\partial}{\partial x^\mu} = \partial_\mu.
$$

You have to accept this. It's just convention to drop the $x$ on the RHS. You're not going to find a deep mathematical reason why there's an $x$ in one and not the other. It's just the way people choose to write things. There's a certain logic in it, of course, but it could have been different.

 

[stevendaryl]

This would be $\partial/\partial x_\mu = \partial^\mu$.

Does $x_\mu$ have a standard meaning? $x^\mu$ means a coordinate, usually, but what is $x_\mu$?

 

[Orodruin]

Does $x_\mu$ have a standard meaning? $x^\mu$ means a coordinate, usually, but what is $x_\mu$?

Typically $x_\mu = g_{\mu\nu}x^\nu$, even if $x^\nu$ is a coordinate and not a vector. This is particularly true in Minkowski space where the concept of a spacetime vector that does have $x^\nu$ as its components (in standard coordinates) does make sense, since it is an affine space (just like a position vector makes sense in Euclidean space). Come to think of it, I would strongly advise against using this notation in anything other than Minkowski space.