Often the actual meaning of the symbols is subordinate to the point you're trying to convey. e.g. I can tell you that `integrate(boundary(Region), form) = integrate(Region, differentiate(form))`, which is great and all, but I might write `<∂M|w> = <M|dw>` because what I'm trying to tell you is that you should think of these things as a dual-pairing of vector spaces (via integration) and that ∂ and d are somehow adjoint. They're both Stokes' theorem, but the emphasis is different, and in either case the hard part is the mountain of work it takes to define what the words even mean (limits, and integrals, and derivatives, and vectors, and covectors, and manifolds, and tangent spaces, and vector fields, and covector fields, and partitions of unity, and symmetric and alternating forms, and exterior derivatives, etc. etc. all so you can finally write one equation, which really just says that all the swirlies inside a region cancel out so if you want to add them all up, you can just add up the outer swirly).

The thing about math is you need to be comfortable viewing the same concept through a bunch of different lenses, and various notations are meant to help you do that by emphasizing different aspects of "the picture" you're looking at.