How does calculus generalize? I am trying to gain an overview of the various generalizations of calculus: what they are, how they relate, and what kind of applications they have. (My knowledge of calculus beyond the standard single/multi variable calculus is pretty limited.)
I am interested in any resources that can give me the broad strokes of what's out there beyond regular calculus. Below I will list some generalizations that I am aware of.

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*Analysis: This is, roughly speaking, rigorous calculus.

*Measure theory: A subset of analysis, this focuses a lot on various types of integrals, which are often better in some sense than the regular calculus integral. (Barry Simon refers to measure theory as "the consummate integral calculus."*)

*Distribution theory: I don't know much about these (according to Wikipedia, distributions are generalizations of functions). Barry Simon describes distribution theory as "the ultimate differential calculus."

*Gauge theory: I have seen gauge theory described as derivatives with respect to a custom reference. The idea is that regular differentiation gives a slope relative to zero, whereas in gauge theory, you can get a slope relative to another slope.

*Differential geometry: This is a generalization of calculus at least in the case of Stokes' theorem, which unifies various theorems in single and multivariable calculus.

There may be more generalizations that I'm unaware of. Ideally I am not just looking for a list of generalizations of calculus; I'm trying to understand how these pieces relate, if they do at all. For example, if distribution theory and gauge theory both generalize the notion of a derivative, are they related at all? If differential geometry and measure theory generalize the notion of an integral, are they related at all? (I think there is a field called geometric measure theory that may be involved here.)
I appreciate any explanations or resources. I searched Google for "generalizations of calculus" but see much that addressed my questions.
*Real Analysis: A Comprehensive Course in Analysis, p. xvii.
 A: A few more:  The weak derivative generalizes the notion of derivative,  it basically looks for functions that would work in the integration by parts formula as the derivative against all suitable test functions https://en.wikipedia.org/wiki/Weak_derivative , that leads to https://en.wikipedia.org/wiki/Sobolev_space , where we get more solutions to differential equations
There are a ton of generalizations of integrals that are used in different applications:  https://en.wikipedia.org/wiki/Integral#Other_integrals
A: Analysis and especially modern mathematics are huge topics, so it's basically impossible to write down every topic related to mathematics. Indeed, math is an interconnected subject, so almost everything is related to each other in some way! Nonetheless, let me try to describe some topics in which analysis is extensively used (or usually seen as "belonging to analysis").
Let me also say that I wouldn't say that any of these things are "generalizations" but rather that they all analytical topics and calculus is just the beginner version of it.

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*Calculus: I believe this term is primarily used in english-speaking countries such as the US. E.g. in Germany, there is no course such as calculus and there is also no proper translation for the word calculus! It is usually seen as a first encounter with basic notions like limits, derivatives, integrals and heavy on the computational aspects of these (product rule, chain rule, substitution by parts, change of basis, ...).

*Multivariable calculus: It is just what the name suggests. The same thing but with more than one variable. In multiple dimensions a lot more can happen and the fact that derivatives are really linear maps (instead of just numbers, as one-dimensional calculus might suggest) becomes more apparent.

*Real analysis: A more rigorous study of the above subjects, more concentrated on the analytical properties and less on computation.

*Complex analysis: All of these names are quite suggestive, eh? Analysis but with a complex variable! Surprisingly, complex analysis is a LOT nicer than real analysis, e.g. all differentiable functions are infinitely often differentiable (even analytic), there are no non-trivial bounded functions, integrals are completely determined by some simply connected set, etc. In his book, Charles Pugh writes


"Complex analysis is the good twin and real analysis the evil one:
beautiful formulas and elegant theorems seem to blossom spontaneously
in the complex domain, while toil and pathology rule the reals."


*

*Vector analysis: More or less the study of vector-valued functions with the differential operators $\operatorname{div}, \operatorname{curl}, \nabla, \Delta$. Central topics would be a rigorous treatment of integral theorems like those from Gauß, Green, Stokes.

*Measure theory: A study of so-called measures which rigorously treats the notion of length and volumes. One central question is how to measure areas of arbitrary subsets of the plane $\mathbb{R}^2$ (or more generally of $\mathbb{R}^n$). It turns out that this cannot be done satisfactorily on all subsets but that's not too bad because it is possible on all sets that one can write down. This will be called the Lebesgue measure $\lambda$. Naturally, a treatment of volumes will lead to the notion of integrals, the Lebesgue integral $\int - \, \mathrm{d} \lambda$ being a central notion and a better behaved notion than that of Riemann integrals.

*Probability theory: OK, you might wonder, what does probability theory have to do with analysis? Indeed, if you take more advanced classes on probability theory, then it is basically an analysis course. In probability theory you study the measures $P(-)$ and so with this the subject turns into measure theory. Central notions like expected values and variances are all defined in terms of Lebesgue integrals and you will study the analysis of probability measures.

*Differential geometry: In the elementary sense really a study of geometry notions like curvatures, tangents, normals, ... The central object in modern differential geometry is that of a manifold which is some geometric object that locally looks like the Euclidean space. A prime example would be the earth we're living on, and as you can see, flat Earthers are often confused that this is just a local behaviour of manifolds and want to conclude that the Earth is globally the Euclidean space! So one study locally $\mathbb{R}^n$-regions and thus central notions are tangent planes on which of course derivatives play a crucial role. Hence, e.g. the study of differential forms and Stokes' theorem.

*Complex geometry: Really a combination of (higher) complex analysis, differential geometry and algebraic geometry. One considers complex manifolds and hence adheres to techniques from complex analysis but because morphisms in complex analysis are so well-behaved, they are quite similar to polynomials and the study in algebraic geometry (see also GAGA).

*Functional analysis: A study of infinite-dimensional vector spaces. One consequence of infinite dimension is that the norms can be different and so one really studies normed spaces and their convergence relations. This leads to notions like weak derivatives, Sobolev spaces, weak convergence, weak topology, ... and one studies the objects $\ell^p, L^p$.

*Harmonic analysis: Really a generalization of Fourier analysis where one is interested in Fourier transforms and the Fourier series.

*Differential equations: Different kinds of differential equations describe much that happens in nature and the name already suggests that you are trying to solve equations where derivatives are involved! This includes parts like calculus of variations.

*Non-standard analysis: Maybe you have been bothered that $\mathrm{d}x$ are not numbers but yet they seem to behave like numbers. While differential form already gives one way of making this precise, non-standard analysis is another subject that focuses on this and introduces infinitesimal numbers.

*$p$-adic analysis: In real analysis, one works with the Euclidean norm and their convergence properties but why work with this one? If one endows $\mathbb{Q}$ with the $p$-adic absolute value $|-|_p$, which measures the multiplicity of a prime $p$ in a number, then one enters the realm of $p$-adic analysis and number theory. The analytical and topological properties here can seem quite weird at first, e.g. every point inside an open ball is the center of that ball!

*Analytical number theory: Analytical methods can be used to obtain properties of integers! Prominent theorems would be Gauß' prime number theorem, Dirichlet's theorem on arithmetic progressions with Dirichlet densities, the study of different kinds of $\zeta$-functions, ...

But this is really just a sneak peak of some of the subjects that use analytical methods in a substantial way. If you look for research papers, you will see that there are myriads of subfields of every field of mathematics. I myself am interested in algebra and still need analysis!
All in all though, I don't think you benefit too much reading about what these topics are really about. You can only start getting a feeling for them by studying these subjects yourself, and maybe the above was interesting enough to motivate you to go pick up a book and start reading!
