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- 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?

##

\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?

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