Differential geometry MOC

Smooth field

An -field on a base -manifold ¹ is some -section of a fibre bundle

The space of fields is thus the space of sections .² diff This generalizes

• A scalar field is an -field, which is equivalently viewed as a smooth function ;

• A (tangent) vector field is an -field which is equivalently viewed as a derivation on the algebra of scalar fields, or as assigning a tangent vector to each point in ;

• A (tangent) tensor field is an -field, which is equivalently viewed as a -multilinear map eating vector fields and covector fields.

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develop | en | sembr

Footnotes

1. Usually called spacetime. ↩

2. 2024. Lagrangian field theory, definition 1.1.1, p. 6. ↩