Greek letter phi (φ), in the form that has a tall stroke. It may be either upright or slanted, according to the overall design. Here it looks like the intent is that the slanted ones are italic (as mathematical symbols often are) and the non-slanted ones regular (because they are a symbol that would normally be italic, within text that is as a whole italicized).

... and the non-slanted ones regular (because they are a symbol that would normally be italic, within text that is as a whole italicized).

Which is really odd. It's true that, in terms of typographic operators, italic² = regular.

But I've never seen this convention applied to formulae. Variables and function names are usually set in italics no matter the context (the s is, for example). This matters, because some functions and variables are distinguished from others by setting them upright — in quantum mechanics, for example, Schrödinger's wave function is often written as Ψ in its time-dependent, and ψ in its time-independent form (at least in the textbooks I've come across).

As such, I would suggest that you keep all φ's in italic here: both along the axes and in the caption.

But I've never seen this convention applied to formulae.

Yes, it's definitely weird. But I couldn't think of any other reason why it would be vertical in the caption and slanted elsewhere (apart from the possibility that this was completely inadvertent).

I think it only looks different because it's what was available in the context of the typewriter face.

A good face for math will have a distinct closed phi symbol ϕ at Ux03D5 (this is "phi1" in older encodings) to distinguish it from an open phi in the alphabet at Ux03C6.

I think it only looks different because it's what was available in the context of the typewriter face.

It's not about the shape, it's about the fact that an upright symbol was used in an italic context, thus following a convention that doesn't apply to maths (even though apparently the italic phi was available).

On second thought, I'd say that setting captions in italics is a fairly stupid idea here. Formulae rely on the differences between italics (variables/functions), roman (units) and bolds (vectors). Changing the context like this is just going to make things confusing.

Formulas should be the same, whether in an italic context (theorems are normally set in italics) or an upright context (definitions are normally set upright). If a letter phi represents an angle, then it should look absolutely the same inside a definition or inside a theorem. In the example above, not only the phi was changed, but also the s. You just don't do that.

The conventions for the formulas are not universal. French mathematics uses upright roman capitals and upright greek letters (independently of the case). In the US, latin capitals and small greek letters are in italics, but greek capitals are normally upright. Journals may have their preferences for the font to be used for vectors, tensors, etc. Once you are given the conventions, you stick to them. If a symbol is to be written with an upright bold sans, then it needs to be an upright bold sans everywhere in the text, independently of the context.

[Added] Here is how the formula may look like with the LaTeX Fourier math font (Utopia based).

The parentheses are upright even when the letters are in italics. The first line uses the US convention, the second uses the French convention. The examples show both LaTeX symbols \phi and \varphi.

I agree that it looks horrible by modern standards, but I believe it's the result of the limited resources of a bygone time. Someone probably pasted the diagrams as images from another source, and the manuscript and captions probably come from an IBM Selectric typewriter (note the paragraph about mathematics in the Wikipedia article). That's why Luca has all this work to do in the first place. I notice that the phi does have a mild slant even though it disappears in comparison to the slant of Courier.

the manuscript and captions probably come from an IBM Selectric typewriter

The phi in the caption indeed matches one probably coming from a Selectric typewriter. Here is another one, coming from my own thesis (also typed on a Selectric).

Once you get used to the style, you don't find it that horrible, provided the notations are consistent.

French mathematics uses upright roman capitals and upright greek letters (independently of the case).

That's really interesting, thanks!

The (experimental) unicode-math XeLaTeX package (pdf doc, 332K) has four math-style options: ISO, TeX, french and upright. Here is a grab from the above pdf.

Interesting!

I'm only wondering how you'd handle units then — grams and g's, Volumes and Volts? Or is it the other way around? And do most French journals still adhere to this tradition?

PS. For next Christmas, I'd like to get TeX rendering on typophile.

I don't know for physics. There are interesting slides by Ulrik Vieth (pdf, 608K) that provide some useful information for various conventions in mathematical physics in general. There is also a nice study of the Russian tradition for mathematics (pdf, 200K) but I have no idea up to what extent mathematical physics may depart from it.

I've seen technical journals from the 1980s that set variables and inline equations all upright and displayed equations in italics. I found this quite distracting, and am glad it has gone away.

See this post concerning a Beautiful Universe font for my physics research. (This illustration below from my paper also uses Φ usually used for electrical or gravitational potential).