Is the Gradient of a Contravariant Vector a Covariant Vector?

[nigelscott]

I have:

dV^μ = (∂V^μ/∂x^η)dx^η where V^μ is a contravariant vector field

I believe the () term on the RHS is a covariant tensor. Is the dot product of () and dx^η a scalar and how do I write this is compact form. I know how this works for scalars but am not clear when tensors are involved.

 

[Ben Niehoff]

I have:

dV^μ = (∂V^μ/∂x^η)dx^η where V^μ is a contravariant vector field

I believe the () term on the RHS is a covariant tensor. Is the dot product of () and dx^η a scalar and how do I write this is compact form. I know how this works for scalars but am not clear when tensors are involved.

You haven't been very specific. What kind of space are you in? Is it flat or curved?

Generically speaking, $\partial_\nu V^\mu$ is not a tensor at all, because in curved space, partial derivatives do not transform nicely under changes of coordinates. Using a covariant derivative,

\nabla_\nu V^\mu
is a tensor of mixed type.

In curved space, "$\mathrm{d} V^\mu$" is not really a sensible thing to do, because it is not covariant under coordinate changes.

 

[nigelscott]

Flat space. I am trying to show that the gradient of a contravariant vector is a covariant vector. I understand how to show this for a scalar, but not sure how to extend this to vectors/tensors.