My sense of this mapping is that it only exists for a small subset of mathematics. It might be the case that you're only interested in learning math insofar as it has practical applications (relevant to some technical field), but most higher-level mathematics (topology, analysis, algebra, logic) have an unclear relation to technical fields, i.e. there are fields that use concepts drawn from those subject areas but have no 1-1 connection as there might be for mechanics/calculus.
I'm not aware of any such mapping, but just off the top of my head:
economics: calculus, optimization problems, probability theory
computer science: computability theory, complexity theory
physics: calculus, differential equations
It might also be useful to go through a course catalog to familiarize yourself with the existence of various technical fields/mathematics. You might be able to explicitly trace dependencies to figure out what the connected fields are, but I don't expect this to work in general. Caltech's course catalog can be located: catalog.caltech.edu/documents/3199/caltech_catalog-1819.pdf
My sense is that if you want to learn math so you can do stuff with it, you should learn slightly more than the minimum amount required to be able to do what you want. Anything else is probably wasted effort. I consider most of the value of the amount of math I've learned to be in making my brain into a different shape and giving me more abstractions than any concrete knowledge I now possess; it's not clear to me what I can do now that I couldn't do before, but it is clear that I can think about problems from more angles.
If it is interesting to you, the level hierarchy for mathematics in my brain looks something like:
Level 1: calculus, probability/statistics, linear algebra, number theory
Level 1.5: differential equations [technically a subfield of calculus, but it's usually split off because it involves a bunch of techniques] [aka an entire class on "how to solve fewer differential equations using more time than Mathematica"]
Level 2: abstract algebra, analysis [with probability as a subfield], topology, logic + intuitive set theory (model theory?) [technically, this contains computability theory and complexity theory as a subfield, but it's pretty independent] [technically, category theory also lives here]
Level 3: algebraic topology, algebraic number theory, descriptive set theory, even abstracter algebra, topology but like weirder, logic but you chart the implications of various assumptions in excruciating detail, <things that combine 2 or more things for level 2>, <things that build on level 2 things more deeply>
Level 4: this is too far above my current level to even figure out what's happening. I have a fairly strong belief that the value of math starts to rapidly diminish the higher the "level" and that this level is almost completely aesthetic.