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I am reading a paper and trying to understand a calculation and all of a sudden I bump into the following term:

$D^2_y p(t,y(t,x))(\partial_t y(t,x),y^\epsilon(t,x))$

where $p$ is a scalar field (pressure), $D_y^2p$ is the Hessian matrix of $p$ wrt the spacial variables, $y$ and $y^\epsilon$ are vector fields, and $\partial_t y$ is the partial derivative of $y$ wrt $t$.

Now, for understanding the calculations I am assuming that the brackets are notation for:

$\left(D^2_y p(t,y(t,x))\cdot\partial_t y(t,x) \right) \cdot y^\epsilon(t,x))$

This makes sense in many ways, specially since this term ought to be a scalar.

I think I get a confirmation further on in the paper where it is stated that

$D^2_y p (t,y) \left((v \cdot \nabla )y, y^\epsilon-y \right)=\sum_{ijk} \partial^2_{ij}p(t,y) \: v_k \: \partial_k y_i \: (y^\epsilon-y)_j$,

here $v$ is a (velocity) vector field.

My question is: has anybody seen something like this before? Maybe its a notation physicists use. I have never seen it before and I couldn't find it anywhere. So far I couldn't think of any other interpretation (nor I think any other is possible). However, I am having trouble following the author's calculations and maybe this could be the source of my problem.

By the way, the paper I am referring to is the following one: http://math1.unice.fr/~brenier/fichiers.ps.pageperso/cms-semigeo-xz-rigorous.pdf

Thank you very much in advance!

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    I wasn't aware if it made any difference or not--I do not understand these equations. However, I provided the full thing in hopes of making things more easily referenced.2012-04-19

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I believe the Hessian is being treated as a bilinear form; see for instance here, here and here.

If so, your interpretation in terms of products is correct (though it's slightly unusual to denote the product of the matrix $D^2_yp$ and the vector $\partial_t y$ with a dot).

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    @mathscat: Neither had I (and I'm a physicist). You're welcome.2012-04-20