I am aware of the so-called Nanson's formula which relates area elements in a reference configuration to those in a deformed configuration through the adjugate of the jacobian matrix. In Chadwick's Continuum Mechanics another way is offered by differential forms which states (1.38): $$A^*(dx \wedge dy) = (Adx \wedge Ady) \ \ \forall dx,dy $$ where $A^*= \text{det}(A)A^{-1} $ is the adjugate.
I am scrambling to try and learn about exterior calculus (which I have only just discovered), and I would like to know why this is so. I found in Agricola, Friedrich, Forrest, Global Analysis: Differential Forms in Analysis, Geometry, and Physics that on Definition 5 and Theorem 1, pg 14, that the factor of the determinant comes into play.
For $f:U_1\to U_2$ differentiable, and $U_1 \subset \mathbb{R}^n$, $U_2 \subset \mathbb{R}^m$ the pullback of bases of differential forms on $U_2$ through components $f^i$ is given by: $$f^*(dx^i)=\sum_j^n \frac{\partial f^i}{\partial x^j} dx^j $$ Then for $g:U_2\to \mathbb{R}$, and $n=m$ then $$f^*(g\cdot dx^1\wedge \dots \wedge dx^n) = (g\circ f )\cdot \det Df \cdot dx^1\wedge \dots \wedge dx^n $$
I didn't see how to link these ideas yet. I'm not even sure if it is related.
Any tips about where the first equation arises without resorting to Hodge star operations would be great, thanks.