# 17: Appendix F- The Cauchy Stress Tensor

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.