Date: 2024-05-19 | Article ID: 70621 |

The concept of a **gauge field** in classical physics is based on the mathematical concept of a **principal bundle**. Some other classical fields that interact with gauge fields are based on the concept of **vector bundles** that are **associated** with the principal bundle. This article introduces principal bundles and vector bundles as specializations of the general concept of a **fiber bundle**. This is a prerequisite for article 76708, which explains how these things relate to the concept of a gauge field.

**2024-05-19**(added another citation to an already-cited reference)**2024-05-02**(fixed a sentence that conflated local trivialization functions and transition functions in the section about constructing a fiber bundle from trivial patches)**2024-04-20**(reworded the comments about generalizing to infinite-dimensional spaces, added the condition that transition functions should be restricted to a Lie group of diffeomorphisms of the fiber, added a footnote about the condition for the bundle to be trivial, fixed a broken sentence in the section about two meanings of "structure group", added a reference for the definition of local section)**2024-04-01**(fixed the wording of one phrase in the definition of principal G-bundle to clarify that the G-action on the fiber must be free and transitive)**2024-03-07**(added the word "smooth" in two sentences in the chapter about sections compared to fiber-valued functions)**2024-03-04**(updated publication info for one reference, added a footnote about distinguishing between data and conditions)**2024-02-25**(fixed incorrect grammar: "whether or not" -> "whether" when the outcome is in question)**2024-02-25**(tweaked the font used to denote real projective space)**2023-10-26**(deleted a stray subscript, deleted a stray parenthesis, added some vocabulary)**2023-10-05**(corrected a statement in a caption about the significance of the dashed lines, fixed a notation discrepancy in a figure, added a footnote about line bundles, added a reference)**2023-10-01**(first version)

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updated 2024-07-11 |