When designing observers for nonlinear systems, the dynamics of the given system and of the designed observer are usually not expressed in the same coordinates or even have states evolving in different spaces. In general, the function, denoted tau (or its inverse, denoted tau*) giving one state in terms of the other is not explicitly known and this creates implementation issues. We propose to get around this problem by expressing the observer dynamics in the the same coordinates as the given system. But this may force us to add extra coordinates, a problem that we call augmentation. This may also force us to modify the domain or the range of the "augmented" tau or tau*, a problem that we call extension. We show that the augmentation problem can be solved partly by a continuous completion of a free family of vectors and that the extension problem can be solved by a function extension making the image of the extended function the whole space. We also show how augmentation and extension can be done without modifying the observer dynamics and, therefore, with maintaining convergence. Several examples illustrate our results.