Talk:Tensor derivative (continuum mechanics)

References present but no citation

There are references here and I'm sure that they check out, the proofs look good but could someone cite the proof (i.e. from which chapter of which books can we find these proofs?).

Inconsistency in the gradient definition

The text says:

The gradient, ${\boldsymbol {\nabla }}{\boldsymbol {T}}$, of a tensor field ${\boldsymbol {T}}(\mathbf {x} )$ in the direction of an arbitrary constant vector $\mathbf {c}$ is defined as:

${\boldsymbol {\nabla }}{\boldsymbol {T}}\cdot \mathbf {c} =\left.{\cfrac {d}{d\alpha }}~{\boldsymbol {T}}(\mathbf {x} +\alpha \mathbf {c} )\right|_{\alpha =0}$

This expression assumes that a dot product exists between the gradient and the constant vector c. Therefore the gradient of a generic tensor should always be a vector. But then the text says:

The gradient of a tensor field of order $n$ is a tensor field of order $n+1$.

I assume therefore that the given definition is not the general one. Am I missing something?

--Juansempere (talk) 18:54, 10 October 2012 (UTC)

The gradient will be a tensor field of rank n+1, and then when you dot with a vector you reduce the rank again by one, leaving a tensor field of rank n. So this checks out.

If you're concerned about the dot product being well-defined, the definition of the dot product between a tensor and a vector is given in coordinates:

$(\mathbf {T} \cdot \mathbf {v} )_{ijkl}=T_{ijk...lm}v_{m}$

129.32.11.206 (talk) 16:59, 11 October 2012 (UTC)