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are these fields separate from each other?
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You are currently reading a thread in /sci/ - Science & Math

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are these fields separate from each other?

>calculus
>linear algebra
>differential equations
>analysis

meaning, do you not need one for the other? are they all independent pathways?
>>
>>7842190
calculus and differential equations preceded analysis. Analysis is a more rigorous "version" of calculus and other theories(complex, functional anaylsis) and is consider more fundamental while calculus is more about application and function.

Linear Algebra is quite a different subject but has applications within differential equations
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>>7842190
>>7842193
Oh, and you usually take calc first, then diffy Qs or linear algebra after your 2nd semester of calc. If you want to take a real analysis course you need an introduction to proofs. There is also sometimes a more advance differential equations course aimed for mathematicians instead of engineers/scientist witch requires knowledge of proofs and linear algebra as well
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>>7842193
i was told that in order to study differential equations you needed to know functional analysis. is there any weight to this?
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>>7842202
At the graduate level functional analysis is very useful for studying different equations, as a linear differential equation can be viewed as an operator on a topological vector space.
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>>7842202
Not really.

No university teaches functional analysis before ordinary differential equations. Some universities won't even ask you to do functional analysis at all, but you can do differential equations in like 2nd year.

Partial differential equations can use methods from functional analysis but partial differential equations are a much richer and more advanced topic.
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>>7842202
Maybe for higher level courses but for most an introduction all you need is two semesters of calc.
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>>7842204
if i wanted to study multivariable calc, algebraic structures, differential eqs both ordinary and partia, and analysis, should i split each up according to calculus, linear algebra, differential eqs, and intro to proofs for the 1st year and then take the 2000 level courses in the 2nd?
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Continuing from>>7842204

I forgot to mention that functional analysis is most useful for proving existence of solutions and neighbourhoof of convergence etc. For actual useful stuff like finding a closed form solution it's not very useful. That's what numerical analysis is for.
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>>7842214
my university shows ODE with numerical methods and then i can opt for an alternative called ODE with Laplace Transform. is that a common distinction?
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>>7842213
-Do 3 courses of calculus (differential, integral, multivariable)
-2 courses of linear algebra (matrix computation and then theory of vector spaces).
-2 courses in ODEs (Basic finding solutions, and then manifold vector field based)
-1 course in PDEs (Fourier series solutions for heat, potential and wave equation)
-2 courses of computer science (procedural and then object-oriented)
-2 courses in numerical methods (Numerical linear algebra and then numerical integration/ differentiation)
After you've done this, you have the basics of applied mathematics for an undergrad. If you want to go further you will need:
- 1 course in discrete math/ proof theory
- 3 courses in analysis (Real analysis, differential geometry, complex analysis)
- Analysis-based modern ODEs

At this point you're at a level where you can start doing research, and take other topics as they apply (functional analysis, measure theory, lie algebra, etc). It ultimately comes down to what your goals are.
>>
where does abstract algebra fit into this
>>
>>7842213
Usually a math student takes calc 1 and 2 as soon as possible, often getting credit from before uni if possible. They then go to take a proof class as soon as possible. Then they have more options witch to take after that, their usually is two linear algebra courses you need to take, and to take an abstract algebra you need proofs and at least on course in linear algebra. Usually its good to take calc III(multivariable calc),proofs and your first linear algebra course around your sophomore year unless you brought in ap calc or something)

You can also take differential equations after taking two calc classes, my university has three options for a diff equations option, a basic class aimed at engineers that only requires calc 2, a more advance one that requires linear algebra and proofs and dynamical systems with the same prereqs. Only one of them can count as credit toward a major/minor but all any of them will let you take PDEs. It might be little bit different at different schools,
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>>7842229
abstract algebra is a huge field, most of what we've been aimed at someones who more interested in anaylsis >>7842224 mentions taking PDEs, multiple ODE courses, and three analysis course but this isn't necessary for all math majors. Usually all you need is two analysis courses and your calc courses. But you can use analysis courses to fill out credit requirements. Abstract algebra is taken after your have one linear algebra course and knowledge of proofs. You usually will take a sequence of two abstract algebra courses in your junior/senior year. If you want more classes in that topic you usually need to take grad classes or a seminar or something of that sort, but just two courses is all you need for all most all classes
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>>7842190

Calculus (babby vocab and grammar up through 5th grade) and differential equations (a set of techniques and models now that you know the language) are two branches of what is more broadly called Analysis (the whole area, overall). Those three are all "the same thing" in the sense that they treat of and are concerned with fundamentally like objects - derivatives, integrals, the continuum.

The one that "doesn't belong" is linear algebra, but its motivations are simple enough that it is amenable to introduction at about the same time as calculus, whenever that might be in an education. Linear algebra is in some (this is just me talking, now) sense /discrete/ (even the mental process of cataloguing its results is in some qualitative sense more discrete), and concerned with answering very different types of questions about dimension. The notion of linear independence of vectors is a rather different inquiry from the notion of what a limit is. It is this vague, qualitative sense of discreteness I refer to which puts linear algebra alongside the discrete 'pops' of (general) algebra, and for that matter discrete math.

But of course, once you know the one thing, you can blend it with the other.
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>>7842190
Calculus is a subset and application of analysis.

Analysis could essentially be summed up as the study of all things related to infinity and is in itself an application of topology.

Differential equations is just a good application of analysis which builds on the application of calculus.

Linear algebra is a subset of abstract algebra, which is basically just the study of mathematical objects and their properties.

If you wanted to learn all of these, note that calculus and linear algebra could be learned independently without much overlap. Analysis could technically be learned independently of all of these as it contains calculus and differential equations in itself, but this would not be advised because the general study of analysis approaches these studies in a more rigorous way. Differential equations would require an understanding of calculus to learn and that is it.

Hope this helps.
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>>7842280
>Analysis could essentially be summed up as the study of all things related to infinity and is in itself an application of topology.

Mostly true but measure theory can be done without a topology
>>
Op here. Is spectral theory a branch of analysis but without the restriction of dimensions?
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