Vector and Tensor Formulation of Fundamentals

Co-ordinate systems:

Let (x${}_{{1}}$, x${}_{{2}}$, x${}_{{3}})$ be a rectilinear, orthogonal coordinate system. The vector components of a point P are given by $\vec{R}=\mathop{OP}\limits^{\to}=[x_{1},x_{2},x_{3}]$.

Let (u${}_{{1}}$, u${}_{{2}}$, u${}_{{3}})$ be a curvilinear, orthogonal coordinate system. The co-ordinate surfaces are given by

$$\begin{array}[]{@{}l@{\,}c@{\,}l}u_{1}(x_{1},x_{2},x_{3})&=&const,\\ u_{2}(x_{1},x_{2},x_{3})&=&const,\\ u_{3}(x_{1},x_{2},x_{3})&=&const.\\ \end{array}$$

The intersection of two co-ordinate surfaces is a co-ordinate line.

Tangent vectors at co-ordinate lines:

$$\begin{array}[]{@{}l@{\,}c@{\,}l}\vec{R}_{1}&=&\displaystyle\frac{\partial\vec{R}}{\partial u_{1}}=\left[{\frac{\partial x_{1}}{\partial u_{1}}\;,\;\frac{\partial x_{2}}{\partial u_{1}}\;,\;\frac{\partial x_{3}}{\partial u_{1}}}\right]\,,\\ \vec{R}_{2}&=&\displaystyle\frac{\partial\vec{R}}{\partial u_{2}}=\left[{\frac{\partial...