Fundamentals of Continuum Mechanics

2.6 Curvilinear coordinates

Recall that any basis for the three-dimensional inner product space ${\epsilon }^3$ has three elements. Also recall that ${\epsilon }^3$ has a fixed orthonormal basis {e_i} = {e₁, e₂, e₃}. As was demonstrated in (2.21), any vector u in ${\epsilon }^3$ can be expressed in terms of this basis, i.e.,

$\mathbf{u}=u_1{\mathbf{e}}_1+u_2{\mathbf{e}}_2+u_3{\mathbf{e}}_3,$

or, compactly,

$\mathbf{u}=u_i{\mathbf{e}}_i.$

The scalars u₁, u