Electroosmotic flow is motion produced by the action of an applied electric field on a fluid with a net charge. At most fluid-solid interfaces, a Debye layer of mobile net charge exists that is typically 1 to 100 nm thick. An applied electric field of O(10 V/mm) produces O(1 mm/s) flow velocities irrespective of flow channel size down to the Debye length scale. Electroosmosis has important applications for manipulation of fluids and actuation in microdevices and in environmental remediation. We show theoretically that electroosmotic flow of uniform fluids in uniform channels is irrotational and the velocities are everywhere proportional to the local electric field provided the channels are large compared to the Debye length. Hence uniform electroosmosis is an inviscid "potential flow" down to extremely small Reynolds number and through media with as small as nanometer-scale dimensions. Rotational flow and pressure-gradients can be applied or induced by surface or fluid non-uniformity. In a microdevice, rotational flow can be beneficial or detrimental depending upon the device's function. To assist with the diagnosis of microdevices, we have developed a particle-image velocimetry (PIV) methodology that allows high-resolution measurement of velocity fields. We demonstrate the ability to infer pressure-driven and electroosmotic components of the flow velocity, and measure both Lagrangian and Eulerian velocity fields with high accuracy and resolution.