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I was trying to understand what is meant by "area preserving map"?. I was going through the Wolfram article about the area preserving map here but any motivation, intuition or visualization to understand the area-preserving maps, like how it is connected with the determinant being $\pm 1$?, would not it be helpful if we could visualize the area-preserving maps?.

I don't think this soft question about the motivation about the area-preserving maps has been asked before, also will be helpful if one could help with references?. Also, I find it in many research articles.

I could find the following questions on MSE-

*) Non-examples of area preserving map .

*)example of area preserving twist maps

*)area-preserving iff |det|=+1

BAYMAX
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From the functional analysis, informally it says that given two vector spaces $\mathcal{U}$ and $\mathcal{V}$, such that there exists an area-preserving map say $\mathcal{T} \colon \mathcal{U} \rightarrow \mathcal{V}$ then the functional map associated to $\mathcal{T}$ is $\mathcal{C} \in \mathcal{F} ( \mathcal{U}, \mathbb{R}) \rightarrow \mathcal{F} ( \mathcal{V}, \mathbb{R}) $ from function space on $\mathcal{U}$ to function space on $\mathcal{V}$ is orthonormal. Hence determinant of $\mathcal{C}$ is +1 or -1.

M.B.
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