Nicolás Guarín-Zapata
email: nguarinz@eafit.edu.co
github: nicoguaro
February 9, 2017
Two common examples of curvilinear coordinates are
\[\begin{align} x =& \rho \cos\phi\\ y =& \rho \sin\phi\\ z =& z \end{align}\]
\[\begin{align} x =& r \sin\theta \cos\phi\\ y =& r \sin\theta \sin\phi\\ z =& r \cos\theta \end{align}\]
We can write the position vector as
\[d\mathbf{r} = \sum\limits_{i=1}^3 \frac{\partial \mathbf{r}}{\partial u_i} du_i \, .\]
The factor \(\partial \mathbf{r}/\partial u_i\) is a non-unitary vector, we can introduce a normalized based \(\hat{\mathbf{e}}_i\)
\[\frac{\partial \mathbf{r}}{\partial u_i} = h_i \hat{\mathbf{e}}_i \]
where
\[\left| \frac{\partial \mathbf{r}}{\partial u_i}\right| = h_i\]
is the scale factor.
We can rewrite the transformation, in components, as
\[d x_i = \sum_j \frac{\partial x_i}{\partial u_j} du_j = \sum_j J_{ij} du_j \, ,\]
where \(J_{ij} = \partial x_i/\partial u_j\) are the components of the Jacobian matrix.
And, its determinant represents the (local) change in volume of the transformation
\[ |J| = h_1 h_2 h_3\, .\]
We can rewrite the transformation as
\[ d\mathbf{r} = \sum_{i=1} h_i \hat{\mathbf{e}}_i du_i = \sum_{i=1}^3 d\mathbf{l}_i \, ,\]
where \(d\mathbf{l}_i = h_i \hat{\mathbf{e}}_i du_i\) is the line differential along the coordinate \(u_i\).
We can define the surface differentials as the vectors that are perpendicular to the differential areas according to the right hand convention, namely
\[d\mathbf{S}_i = d\mathbf{l}_j \times d\mathbf{l}_k = h_j h_k \hat{\mathbf{e}}_i du_j du_k = \hat{\mathbf{e}}_i dS_i\]
And the volume differential is given by the volume of the curvilinear parallelepiped defined by the line differentials, i.e.
\[dV = d\mathbf{l}_1 \cdot (d\mathbf{l}_2 \times \mathbf{l}_3) = h_1 h_2 h_3 du_1 du_2 du_3 \, .\]
Alonso Sepúlveda Soto. Física matemática. Ciencia y Tecnología. Universidad de Antioquia, 2009.
Wikipedia contributors. "Curvilinear coordinates." Wikipedia, The Free Encyclopedia. Wikipedia, The Free Encyclopedia, Retrieved: 3 Feb. 2017.