/sci/ - Science & Math

[Anonymous No.4380209]

Hey /sci/, I have a Cartesian tensor,
<div class="math">T_{ij}=U_{ij}+V_{ij}+S_{ij}</div>
and <span class="math">V_{ij}[/spoiler] is isotropic, <span class="math">U_{ij}[/spoiler] has zero trace and is symmetric, and <span class="math">S_{ij}[/spoiler] has three independent components.
How do I decompose <span class="math">T_{ij}[/spoiler] into three tensors?

[Anonymous No.4380941]

durrrrr
<div class="math">T_{ij} = \frac{1}{2}(T_{ij} + T{ji}) + \frac{1}{2}(T_{ij} - T_{ji}) \equiv \frac{1}{2}(T_{ij} + T_{ji}) + S_{ij}</div>
so it's obvious S_{ij} has 3 independent components: <span class="math">S_{12},~S_{13},~and~S_{23}.[/spoiler]

you can see that if the trace <span class="math">T_{ii}[/spoiler] is written as <span class="math">T_{0}[/spoiler], then <span class="math">\frac{1}{3} T_{0} \delta_{ij}[/spoiler] is an isotropic tensor <span class="math">V_{ij}[/spoiler]. you get:
<div class="math">TrU_{ij} = \frac{1}{2}(T_{0} + T_{0}) - \frac{1}{3} T_{0}3=0</div>
from symmetry, and so <span class="math">U_{ij}[/spoiler] is traceless and the decomposition is done.