Relation between area elements in finite deformation theory (continuum mechanics) There are relations for the line and volume elements in continuum mechanics. For example:
\begin{align}
\ \ \ \ \ \ \ \ \ \ \ \  \frac{V}{V_0}&={\rm det}(F)\tag{1}\\
\lambda^2&=(F^TFe_1\cdot e_1)\tag{2}
\end{align}
with $F$ being the deformation gradient, 
$$\lambda=\delta x/\delta X \tag{3}$$ 
is the stretch and $e_1$ is the unit vector in direction where stretch is to be found.
Is there a similar relation between infinitesimal areas (for ratio of deformed and undeformed areas)?
 A: Your first two equations can be written in other forms, so how about this?   In 1-D:
$$\lambda=\frac{L}{^0L}$$
(I use a pre-superscript for initial values and post-subscripts for spatial directions).   In 2-D:
$$\lambda_1\lambda_2=\frac{L_1}{^0L_1}\frac{L_2}{^0L_2}=\frac{A}{^0A}$$
In 3-D:
$$\lambda_1\lambda_2\lambda_3=\frac{L_1}{^0L_1}\frac{L_2}{^0L_2}\frac{L_3}{^0L_3}=\frac{V}{^0V}$$
Can't recall ever using the 2-D case but it might arise in a plane stress or plane strain calculation.
A: There is, and it is usually referred to as Nanson's formula/equation.
Consider an area element with initial size $da$ and normal orientation vector $\vec{n}$ that is deformed into an area element of size $dA$ with normal orientation vector $\vec{N}$.
The relationship describing these oriented area elements before/after deformation is:
$$\vec{n}\ da = (\det{\pmb{F}})(\pmb{F}^{-T}\cdot \vec{N})dA$$
where $\pmb{F}$ is the deformation gradient tensor. See A.J.M. Spencer's book on continuum mechanics or this page for a derivation of it.
