I have a 2D manifold (triangle) defined by points $\vec{X}=(X_1, X_2, X_3)$ in the reference configuration and have $X_3=C$ (moreover, the reference configuration is not the unitary simplex ); when I apply the transformation: \begin{align} \vec{x}=\vec{\phi}(\vec{X})=\left\{\begin{matrix} x_1= X_1\\ x_2= X_2\\ x_3=\alpha X_3 \end{matrix}\right. \end{align}
I'm expecting it's deformation gradient to be the unit matrix $\mathrm{I}_{3\times 3}$. However, if I apply derivation of $\vec{\phi}$ with respect to $\vec{X}$, what I obtain is:
\begin{align} F_{ij}=\frac{\partial x_i}{\partial X_j}=\left[\begin{matrix} 1 &0 &0 \\ 0 &1 &0 \\ 0 &0 &\alpha \end{matrix}\right] \end{align}
When I consider a unit simplex, the mapping between the unit simplex and the reference configuration $\vec{X}$ with covariant basis $C=[\vec{A}_1,\vec{A}_2,\vec{A}_3]$ where: $\vec{A}_3=(\vec{A}_1\times \vec{A}_2)/\|(\vec{A}_1\times \vec{A}_2)\|$, and the mapping between the unit simplex and the current configuration $\vec{x}$ with covariant basis $c=[\vec{a}_1,\vec{a}_2,\vec{a}_3]$ where: $\vec{a}_3= (\vec{a}_1\times \vec{a}_2)/\|(\vec{a}_1\times \vec{a}_2)\|$; I obtain as deformation gradient: $F=cC^{-1}=\mathrm{I}_{3\times3}$
Why $F=\frac{\partial x_i}{\partial X_j}$ yields thew wrong deformation gradient, while $F=cC^{-1}$ the right one? How can I obtain the correct deformation gradient deriving $\vec{x}=\vec{\phi}(\vec{X})$?
Thanks