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Right now I am studying dynamical systems where I often encounter the problems of the form:

Find the image of the unit disk $\{(x_1,x_2): x_1^2+x_2^2 \leq R\},R>0$, under the linear flow $\phi=e^{At}$, where $$A=\begin{bmatrix} \lambda_1 & 0 \\ 0 & \lambda_2 \\ \end{bmatrix} \quad \lambda_1<0<\lambda_2 $$ Under what condition will the flow preserve the area of the disk?

The confusing part to me is the line (Under what condition will the flow preserve the area of the disk?) as I don't understand what it means, and it feels like there is a physical interpretation of the problem that I am missing.

Any help or resources to help me understand the problem in the larger context of dynamical systems would be helpful.

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    When you apply $\phi$ to all points in the disc $D$ you obtain a region which has some area. Preserving the area simply means that the area of this set is the same as that of $D$ for all $t$. – 2017-01-22
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    If you compute $\phi \pmatrix{x\\y}$ it will be become clear what is the effect of applying $\phi$ to a general vector (it will correspond to "stretching" the axes by some factor) and thereby for the area. – 2017-01-22
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    Please replace "the flow is preserved" (absurd in the present setting) by "the flow preserves [the area of some domain]". – 2017-01-23

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