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Polar eigenvectors: a better way?

All differential equations students have encountered eigenvectors and eigenvalues in their study of systems of linear differential equations. The eigenvectors and phase plane solutions are displayed in a Cartesian plane, yet a geometric understanding can be enhanced, and is arguably better, if the system is represented in polar coordinates. A consequence of using the polar form is that the geometric characteristics of centres and spirals are immediately identified, while the features of saddles and nodes are also made clear.