Product of dyadic and a vector

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.