The Rotational Trace

We combine ideas from geometry, calculus, complex numbers, and linear algebra to find a simple way to characterize the tendency of solutions of a 2×2 linear system of differential equations to rotate in an averaged sense. This characterization extends to solutions that do not truly rotate by giving geometric information about the eigenvectors or generalized eigenvectors of the corresponding coefficient matrix. Toward these ends, we introduce a scalar quantity that we name the rotational trace. We show that the sign of the rotational trace predicts the direction of rotational motion in an averaged sense. In addition, the rotational trace is related to well-known invariants of 2×2 matrices such as the trace under rotational transformations. Finally, we connect the rotational trace to vector calculus by finding finite approximations that compute the flux and rotation of the system of differential equations exactly while using as few as three points and provide a simple proof of the right-hand rule for cross-products.