Cartesian coordinates

Choose an origin $o$ and an orthonormal basis $(e_1,\ldots,e_n)$ in Euclidean space. The Cartesian coordinates of a point are the unique real numbers satisfying

x=o+\sum_{i=1}^n x_i e_i.

The basis vectors are constant as the point varies. Hence differentiating a vector-valued function in these coordinates differentiates only its component functions.

Comparison with a moving basis

For $u(q)=\sum_i u_i(q)e_i(q)$ in a moving basis, the product rule instead gives $\partial_j u=\sum_i(\partial_j u_i)e_i+u_i\partial_j e_i$ . Those extra terms are essential in cylindrical coordinates. Cartesian components also provide a direct way to check smoothness at points where another coordinate system degenerates.