Matrix-valued tensor field in Euclidean coordinates

In a fixed Euclidean basis, a rank-two tensor field is represented by a matrix-valued map $T:U\to\mathbb R^{n\times n}$ . Here both tensor indices are identified using the Euclidean inner product. Under an orthogonal change of coordinates $x'=Qx$ , its components transform by

T'(x')=QT(x)Q^T.

This transformation rule distinguishes a tensor from an array of scalar fields with no specified geometric meaning.

Momentum flux

For a vector field $u$ , the outer product $u\otimes u$ has entries $u_i u_j$ . Its row divergence is the vector with components $\sum_j\partial_j(u_i u_j)$ . Stating which index is differentiated fixes the convention for nonsymmetric tensors.