# 17: Appendix F- The Cauchy Stress Tensor

- Page ID
- 18044

In section 6.3.2 we defined the Cauchy array, whose elements are the components of the stress vector \(\vec{f}\) acting on each of the three coordinate planes:

\[\tau_{i j}=f_{j}^{(i)}.\]

In this appendix we will demonstrate three additional properties of this array:

- The stress vector acting on
*any*plane is given by \(f_j=\tau_{ij}n_i\), where \(\hat{n}\) is the unit normal to the plane in question. - The array \(\underset{\sim}{\tau}\) transforms as a 2nd-order tensor.
- \(\underset{\sim}{\tau}\) is symmetric.

We will do this by applying Newton’s second law to carefully chosen fluid parcels and imagining the result if we take the size of the parcel to zero.