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In my finite element method book, there is a notation which is confusing me. Given $v:R^2\rightarrow R^2$, I'm supposed to evaluate

$\sigma\cdot \nabla v^T$
where $\sigma$ is a smooth tensor valued function. What is confusing me is the notation $\nabla v^T$. How do I interpret the gradient of a vector? Is it a matrix or a vector? Also, should I interpret the equation as $\nabla (v^T)$ or $\nabla^T v$?

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The gradient of a vector field is

$ (\nabla v)_{ij} = \frac{\partial v_i}{\partial x_j} $

and so

$ \sigma\cdot\nabla v^T=\sigma\cdot(\nabla v)^T=\sum_{i,j=1}^3\sigma_{ij}\frac{\partial v_i}{\partial x_j} $

(given that $A\cdot B=\sum_{ij}A_{ij}B_{ji}$ and so $A\cdot B^T=\sum_{ij}A_{ij}B_{ij}$)

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    It is a inner product in the vector space of tensors, don't know the name in English.2012-09-15