## Curvilinear Coordinates

Suppose we have a curvilinear coordinate system with coordinates
$(u_1 , u_2 , u_3 )$

We can express the ordinary Cartesian coordinates in terms of these curvilinear coordinates.
$x=x(u_1 , u_2 ,u_3 )$

$y=y(u_1 , u_2 ,u_3 )$

$z=z(u_1 , u_2 ,u_3 )$

Hence
$\mathbf{r}=\mathbf{r}(u_1 , u_2 ,u_3 )$

A tangent vector to the
$u_1$

curve is
$\frac{\partial \mathbf{r}}{\partial u_1}$
and a unit vector tangent to the
$u_1$
curve is
$\mathbf{e_1} = \frac{\partial \mathbf{r} / \partial u_1}{ |\partial \mathbf{r} / \partial u_1 |}$

Similarly, unit vectors tangent to the
$u_2 , \; u_3$
curves are
$\mathbf{e_2} = \frac{\partial \mathbf{r} / \partial u_2}{ |\partial \mathbf{r} / \partial u_2 |} , \: \mathbf{e_3} = \frac{\partial \mathbf{r} / \partial u_2}{ |\partial \mathbf{r} / \partial u_3 |}$

A curvilinear system is said to be orthogonal if the coordinates axes meet at right angles. If the vectors
$\mathbf{e_1} , \: \mathbf{e_2} , \: \mathbf{e_3}$
are mutually perpendicular the the system is orthogonal and if the dot product of each is 1, the system is also 1, then the system is orthonormal.
$\mathbf{e_i} \cdot \mathbf{j} = \left\{ \begin{array}{cc} 0 & i \neq j \\ 1 & i=j \end{array} \right.$

The system is orthonirmal
Also the unit tangent vectors
$\mathbf{e_1} , \: \mathbf{e_2} , \: \mathbf{e_3}$
are perpendicular to their respective coordinate surfaces so that the vector
$\mathbf{e_1}$
would be perpendicular to the surface
$u_2 = constant$

Also the system is right handed so that
$\mathbf{e_1} \times \mathbf{e_2} = \mathbf{e_3}$