In physics, are all functions fields?

I really confused if there is a function (mostly in physics, functions represents physical quantities) which is not a field? I feel all functions in physics are fields. Is there any functions which are not fields? I see a lot of questions in stackexchange about functions and fields. But no one nail down the difference between functions and a fields in Physics, other than answers resembling textbook explanations.

This was also one of my biggest questions when first learning this stuff. The terms "function, mapping, transformation, functional, scalar field, vector field, tensor field" etc all seemed to be different. But strictly from a mathematical perspective, they are all functions.

In math, the definition of a function is as follows: a function is a triple of information $$(f,A,B)$$ where $$A,B$$ are (non-empty) sets and $$f:A \to B$$ is a "rule" such that for each "input" $$a \in A$$ there is a well-defined "output", which we call $$f(a) \in B$$ (if you want to be super formal, then a function can be defined as a subset $$\Gamma_f \subset A \times B$$ such that $$(a,\xi), (a,\eta) \in \Gamma_f$$ if and only if $$\xi = \eta$$).

Anyway, for our purposes, the intuitive definition of "function" suffices. Namely, it is a triple of information $$f:A \to B$$, where we call $$A$$ the "domain", $$B$$ the "target space", and $$f$$ the "rule". (Sometimes, we refer to $$f$$ itself as the function)

Now we have various terms like "field", "functional", but really these are all functions according to the definition I gave above. The question you should be asking is "what is the domain and what is the target space of the function".

For example, let me talk about what a vector field is.

A vector field on $$\Bbb{R}^n$$ can be defined as a function $$\xi: \Bbb{R}^n \to \Bbb{R}^n$$.

So, a vector field IS a function, but it's just that the domain and target space are $$\Bbb{R}^n$$.

The more "general and correct" definition of a vector field involves the idea of smooth manifolds. Let $$M$$ be a smooth manifold, $$\pi:TM \to M$$ be the tangent bundle. Then, a vector field on $$M$$ is a map $$\xi:M \to TM$$ such that $$\pi \circ \xi = \text{id}_M$$.

Once again, even in this more general situation, a vector field is still a function. The only difference is that we changed the domain and target space, and we added a slight extra technical condition. So, in general every field (in the physics sense, not the algebra sense) is a function (from one set to another set, such that it satisfies a certain technical condition).

Now, typically in physics, when people use the term "function", they often mean something like a function $$f: \Bbb{R} \to \Bbb{R}$$; i.e in common language, people usually assume that the domain and target space are $$\Bbb{R}$$. Sometimes, the word "scalar field" might be used to describe a function $$\phi: \Bbb{R}^3 \to \Bbb{R}$$. An example is the electrostatic potential: at each point $$(x,y,z) \in \Bbb{R}^3$$, we have a number $$\phi(x,y,z) \in \Bbb{R}$$, which we call "the potential at the point $$(x,y,z)$$".

• Nice! I have to explain the field-function relationship to an audience and I figured it went like this. It's good to get confirmation. Feb 8 at 19:25

Functions are a mathematical construct, they have nothing to do with physics other than the fact that we use them as for their mathematical relevance. They become meaningful whenever physicists give them a physical meaning. Fields are, mathematically, functions but they have a deeper meaning in the physical sense. In physics appear many functions as mathematical entities, some of them have a physical meaning. Some examples could be the generating functional (which is actually a function of fields, so a functional), spherical harmonics which for example pop up in the angular distribution of atomic orbitals, Bessel function which pop-up everywhere and are liked, for example, to the pattern of light coming from a slit, distribution functions appear everywhere in quantum mechanics and are actually a meaningful measurable quantity, and so on.

But saying that "all functions in physics come up as fields" is not so good since you're mixing up a mathematical object with a meaningful physical quantity that comes to be of the form of that specific mathematical object, a function.

• "Functions are a mathematical construct, they have nothing to do with physics..." What about some "physically important" functions, like sinusoidal functions (these almost always come up when we talk about oscillations, which are very physical)?
– user258881
May 24, 2020 at 8:55
• Yes, that's one of my points. Sometimes functions have a physical meaning, and a measurable one. But not all functions are fields in physics. But yes, maybe it's not so clear what I intended to say in the beginning. May 24, 2020 at 8:57
• Alright, for example, if I have a non uniform sheet of material. I am heating it up form some point. I can have values of temperature at different points. Suppose, this temperature field cannot be represented by a common function because of the nonuniformity of the material (thickness, composition etc.). Still can I assume the temperature values as a field? Sorry, I am trying to nail down if there are cases where I can say, this is a field but that is not (more physically) May 24, 2020 at 9:06
• Mathematically speaking, to be a field you need it to be quite smooth. In the real world that surely would be the case but maybe your mathematical construct does not have such smoothness in which case you couldn't, mathematically, call it field. But in physics we make many approximations, so in the physical sense, yes, that would be a temperature field. But you're giving us a specific example in which physicists speak of fields. But I, and even @FakeMod, gave you examples in physics in which appear functions, but not as fields. May 24, 2020 at 9:12

Is there any functions which are not fields?

Yes. For example, in projectile motion the position of the particle $$\mathbf{r}(t)$$ is a function (of time), not a field. Fields describe quantities which exist everywhere, like the electric field $$\mathbf{E}(x,y,z,t)$$. A point particle exists at only one point.

A field is a function of space and time.

it is that simple.

if such a function maps to Vectors, its a Vectorfield, if to tensors, its a ... you get it.

A long time ago, a field in physics was the influence of a source on a test object by a force. Later the QED defined fields even without a source.

Normally I would follow the former definition. There is a source - for example an electrically charged body - this has a field and this field is tested by another charged or polarizable body by attraction, repulsion or torque.

In a broader sense, a heat source also has a field whose characterization could be the gradient for the temperature difference at some distance. Again, a probe (a test object) could be heated at some distance to calculate this field (its gradient). This definition is broader because the probe doesn’t feel a force on it.

In a strict sense only separated electric charges, aligned by their magnetic dipoles subatomic particles and the masses with their gravitation influence are sources for fields.