I am going through a book on Mechanics (Introduction to the mechanics of a continuous medium by Lawrence Malvern, page 132 last section), and it has the following claim at a few places
T is a second-order tensor because it is a linear vector function associating antecedent vectors $\vec{n}$ the image vectors $\frac{d\vec{u}}{dS}$
1) But T has the Cartesian components as $\frac{du_i}{dx_j}$. How does this guarantee linearity. Am I missing something here?
For example $u_1$ could be $x_1^3$, then $\frac{du_1}{dx_1}$ would be $3x_1^2$ (i.e. not linear operation)
2) I thought transformation laws dictate if something can be called a Tensor. Does linearity imply the same? (If so won't any matrix qualify as a Tensor)