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Any field (in the physics sense of the word) is a function.

But is any function a field? It seems that one might construct a function that returns a tuple consisting of $6$, temperature and electric potential. We would not consider that a field, right? So where is the line?

Basically I am looking for a decent definition of the term field.

Džuris
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  • This post talks about what a field is. https://physics.stackexchange.com/q/364358/37364 – mmesser314 Apr 14 '20 at 03:06
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    I've removed a number of comments that were attempting to answer the question and/or responses to them. Please keep in mind that comments should be used for suggesting improvements and requesting clarification on the question, not for answering. – David Z Apr 14 '20 at 05:55
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    Maybe this will help: https://physics.stackexchange.com/q/13157/ – Noone Apr 14 '20 at 09:10

1 Answers1

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The following definition i borrowed from a QFT lecture by Tobias Osborne.

A field $\phi$ is a quantity (e.g. density, space, charge, ...) defined at every point in a manifold $\mathcal M$ (spacetime). So for us the manifold is often $\mathcal M = \mathbb R _t \times \mathbb R ^3 _{\vec{r}} $. The field $\phi$ is then a function

$$ \phi: \mathcal M \rightarrow \mathcal S $$

where $\mathcal S$ is a target-space. For example $\mathcal S = \mathbb R$ which would be a scalar field like temperature. Or $\mathcal S = \mathbb R^3$ would be a vector field like the electric field. Usually a field $\phi$ should be at least two times differentiable.

AlmostClueless
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