= Cauchy's stress theorem
{c}
{title2=$\mathbf t(\mathbf n)=\boldsymbol\sigma\mathbf n$}
<Cauchy's stress theorem> states that local contact <traction> depends linearly on the oriented unit <normal vector>: $t_i(\mathbf n)=\sigma_{ij}n_j$, with $\sigma_{ij}=\mathbf e_i\cdot\mathbf t(\mathbf e_j)$. It follows by balancing <forces> on a shrinking tetrahedron with three coordinate faces and one oblique face. Contact <forces> scale as the face <area>, while bounded <body force> and inertial contributions scale as its <volume>; dividing by the oblique face <area> and shrinking eliminates the latter. The coordinate-face projected <areas> are the oblique face <area> times $n_j$, giving the displayed linear map. Symmetry of the <Cauchy stress tensor> requires the additional balance of <angular momentum> in a continuum without independent couples.
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