Ask your students: "Does this picture, of the most important field of all: Im(dz/z), show an irrotational field?" Most invariably answer incorrectly, demonstrating a complete lack of understanding of derivatives as related to LOCAL properties. The most important lesson to learn is that derivatives mean looking through magnifying lenses. The trace-zero, nonzero field, w/ zero skew symmetric part provides the important view of orthogonal eigenspaces, w/ eigenvalues of same magnitude but opposite signs.