Editing Curvilinear coordinates (section)

===Jacobian of the transformation===

The above [[systems of linear equations]] can be written in matrix form using the Einstein summation convention as
:<math>\cfrac{\partial x_i}{\partial q^k} \mathbf{e}_i = \mathbf{b}_k, \quad \cfrac{\partial q^i}{\partial x_k} \mathbf{b}_i = \mathbf{e}_k</math>.

This [[coefficient matrix]] of the linear system is the [[Jacobian matrix]] (and its inverse) of the transformation. These are the equations that can be used to transform a Cartesian basis into a curvilinear basis, and vice versa.

In three dimensions, the expanded forms of these matrices are
:<math>
\mathbf{J} = \begin{bmatrix}
       \cfrac{\partial x_1}{\partial q^1} & \cfrac{\partial x_1}{\partial q^2} & \cfrac{\partial x_1}{\partial q^3} \\
       \cfrac{\partial x_2}{\partial q^1} & \cfrac{\partial x_2}{\partial q^2} & \cfrac{\partial x_2}{\partial q^3} \\
       \cfrac{\partial x_3}{\partial q^1} & \cfrac{\partial x_3}{\partial q^2} & \cfrac{\partial x_3}{\partial q^3} \\
     \end{bmatrix},\quad
\mathbf{J}^{-1} = \begin{bmatrix}
       \cfrac{\partial q^1}{\partial x_1} & \cfrac{\partial q^1}{\partial x_2} & \cfrac{\partial q^1}{\partial x_3} \\
       \cfrac{\partial q^2}{\partial x_1} & \cfrac{\partial q^2}{\partial x_2} & \cfrac{\partial q^2}{\partial x_3} \\
       \cfrac{\partial q^3}{\partial x_1} & \cfrac{\partial q^3}{\partial x_2} & \cfrac{\partial q^3}{\partial x_3} \\
     \end{bmatrix}
 </math>

In the inverse transformation (second equation system), the unknowns are the curvilinear basis vectors. For any specific location there can only exist one and only one set of basis vectors (else the basis is not well defined at that point). This condition is satisfied if and only if the equation system has a single solution. In [[linear algebra]], a linear equation system has a single solution (non-trivial) only if the determinant of its system matrix is non-zero:
:<math> \det(\mathbf{J}^{-1})  \neq 0</math>
which shows the rationale behind the above requirement concerning the inverse Jacobian determinant.