Showing that the gradient of a scalar field is a covariant vector

[AndersF]

Homework Statement

Prove that the covariant gradient of a scalar field is a covariant vector

Relevant Equations

$\nabla f=\frac{\partial f}{\partial x^{i}} g^{i j} \mathbf{e}_{j}$

In a general coordinate system $\{x^1,..., x^n\}$, the Covariant Gradient of a scalar field $f:\mathbb{R}^n \rightarrow \mathbb{R}$ is given by (using Einstein's notation)

$\nabla f=\frac{\partial f}{\partial x^{i}} g^{i j} \mathbf{e}_{j}$

I'm trying to prove that this covariant gradient $\nabla f$ is indeed a covariant vector. To do so, I'm trying to show that it transforms as a 1-covariant tensor under a change of basis.

Let $C$ be the transition matrix from a basis $\{\mathbf e_i\}$ to a basis $\{\tilde {\mathbf e}_i\}$, that is, $\tilde {\mathbf e}_i= \mathbf e_iC^i_j$.

The covariant derivative increases the contravariant tensor order of the tensor by one unit. Since the partial derivative of a scalar field is indeed a covariant derivative, the object $\frac{\partial f}{\partial x^{i}}$ will therefore be a 1-covariant tensor which I will call $F_i$.

On the other hand, the contraction between the dual metric tensor $g^{ij}$ and $F_i$ will raise the subscript $i$ of $F_i$, and the resulting object will be a 1-contravariant tensor: $g^{ij}F_i\equiv H^j$.

But then, $\frac{\partial f}{\partial x^{i}} g^{i j}=H^j$ will transform as the contravariant components of a contravariant vector $\mathbf{v}=H^j\mathbf{e}_{j}$: $\tilde H^j=(C^{-1})^j_kH^k$, which is just the opposite of what I have to prove...

Where is my mistake? How could this be proved?

 

[PeroK]

I don't recognise your notation. I'm more familiar with things like:
$$g^{ij} = g^{k'l'}\frac{\partial x^i}{\partial x^{k'}}\frac{\partial x^j}{\partial x^{l'}}$$If you do the same for the basis vector and partial derivatives, you can express the components of $\nabla' f$ in terms of the components of $\nabla f$ and derive/confirm the relevant transformation rule.

 

[PeroK]

PS the result should drop out in a couple of lines.

 

[AndersF]

PS the result should drop out in a couple of lines.

I too think that the demonstration should be shorter than what I have tried, but I don't find the problem...

By the way, the components of the transition matrix $(C^i_j)$ of my notation are the terms $\frac{\partial x^i}{\partial x'^j}$ of your notation, aren't they?

 

[PeroK]

I too think that the demonstration should be shorter than what I have tried, but I don't find the problem...

If I understood what you were doing, I'd try to help!

 

[ergospherical]

Homework Statement:: Prove that the covariant gradient of a scalar field is a covariant vector

Examine the chain rule: $\partial / \partial x^j = (\partial x'^i/\partial x^j) \partial / \partial x'^i$. The Jacobian $\partial x'^i/\partial x^j$ is your transition matrix.