Date: 2024-06-23 | Article ID: 00418 |

Tensor fields live on smooth manifolds. The concept of a tensor field is independent of any coordinate system: changing the coordinate system has no effect on the tensor field itself, even though it changes the way the tensor field is represented in terms of coordinates. In contrast, given any smooth rearrangement of the manifold's points (that is, any diffeomorphism from the manifold to itself), a corresponding transformation of tensor fields may be defined in a natural way. This article defines that transformation and calls it a **fieldomorphism**. Unlike a coordinate transformation, a fieldomorphism *does* change the tensor fields. In general relativity, any fieldomorphism of a solution of the equations of motion gives another solution of the equations of motion, a property often called **general covariance**. In general relativity, two solutions related to each other by a fieldomorphism are physically equivalent to each other.

**2024-06-23**(changed the wording in a few places to avoid using the overloaded term "gauge symmetry")**2022-06-11**(updated references to other articles in this series to link to html abstract pages instead of to pdfs. Didn't change the version number/date)**2022-02-20**(changed two equation-references, and removed an extra syllable)**2022-02-18**(first version)

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