# Product of dyadic and a vector

1. May 1, 2014

### nigelscott

I have:

dVμ = (∂Vμ/∂xη)dxη where Vμ is a contravariant vector field

I believe the () term on the RHS is a covariant tensor. Is the dot product of () and dxη 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.

2. May 1, 2014

### Ben Niehoff

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.

3. May 1, 2014

### 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.

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