- #1
paccali
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Homework Statement
Calculate the Divergence of a second-order tensor:
[tex]\sigma _{ij}(x_{i})=\sigma_{0}x_{i}x_{j}[/tex]
Homework Equations
[tex]\bigtriangledown \cdot \sigma_{ij}=\sigma_{ij'i}[/tex]
The Attempt at a Solution
[tex]\sigma_{ij'i}=\frac{\partial }{\partial x_{i}}\cdot\sigma_{0}x_{i}x_{j}[/tex]
[tex]=\sigma_{0}(x_{j})[/tex]
I'm not sure if this is correct. When I put it into a matrix form and calculate the divergence, I seem to get:
[tex]\sigma_{0}\begin{bmatrix}
x_{1}^{2} & x_{1}x_{2} & x_{1}x_{3}\\
x_{1}x_{2} & x_{2}^{2} & x_{2}x_{3}\\
x_{1}x_{3} & x_{2}x_{3} & x_{3}^{2}
\end{bmatrix}[/tex]
[tex]\sigma_{ij'i}=\sigma_{0}\begin{bmatrix}
2x_{1} & x_{2} & x_{3}\\
x_{1} & 2x_{2} & x_{3}\\
x_{1} & x_{2} & 2x_{3}
\end{bmatrix}[/tex]
Which doesn't equal the partial that wasn't put into matrix form. Any help?