The motion of an incompressible fluid is a geodesic in the group of volume-preserving diffeomorphisms. Stability of the fluid motion is thus related to growth of Jacobi fields. I will summarize what is known about the sign of the curvature of the diffeomorphism group. I will present the Jacobi equation, and show how to solve it in some very simple examples. These examples illustrate how the sign of the curvature can fail to predict the growth of Jacobi fields; for example, nonpositive curvature need not imply exponential growth. The talk will be based on a recent paper: 'Growth rates of Jacobi fields and Lagrangian stability of anincompressible fluid, part I,' available on my web site