Smooth field

An $𝐹$ -field on a base $𝐶 𝛼$ -manifold $𝑀$ ¹ is some $𝐶 𝛼$ -section of a fibre bundle

𝐹 \to 𝐸 ↠ 𝑀

The space of fields is thus the space of sections $Γ 𝛼 (𝑀, 𝐵)$ .² diff This generalizes

A scalar field $𝜑 \in Γ 𝛼 (𝑀, 𝑀 \times ℝ) ≅ 𝐶 𝛼 (𝑀)$ is an $ℝ$ -field, which is equivalently viewed as a smooth function $𝑀 \to ℝ$ ;
A (tangent) vector field $Φ \in Γ 𝛼 (𝑀, 𝑇 𝑀) ≅ 𝔡 𝔢 𝔯 (𝐶 𝛼 (𝑀))$ is an $ℝ 𝑛$ -field which is equivalently viewed as a derivation on the algebra of scalar fields, or as assigning a tangent vector to each point in $𝑀$ ;
A (tangent) tensor field $𝑇 \in Γ 𝛼 (𝑀, 𝑇 𝑝 𝑞 𝑀)$ is an $𝑇 𝑝 𝑞 ℝ 𝑛$ -field, which is equivalently viewed as a $𝐶 𝛼 (𝑀)$ -multilinear map eating vector fields and covector fields.

Footnotes