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As a high school student I have already completed the equivalent of Calculus 2, covering advanced material such as differential equations and L'hopital's rule, so I have seen a lot of math already. I am also going to take Calculus 3 in college. It seems that the hardest thing in math is integrating complicated integrals. I suppose what I really want to ask is, how much math is there left to learn beyond calculus? And are they on the same level of difficulty as Calculus 3 or harder?

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    Try reading some group theory :P – Kenny Lau Aug 10 '18 at 10:28
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    Some of the smartest people in the world struggle for years to solve mathematical problems, often without success. So yes, there are things that are more difficult than calculus 3 :P. – Eff Aug 10 '18 at 10:28
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    When I finished high-school I felt I knew 50% of maths, when I took some grad courses I felt like I knew 0.01% of it. That's how it was for me. Simply the more you learn you realize you don't know much more – Stefan4024 Aug 10 '18 at 10:34
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    This looks like an honest question, but honestly, it's almost impossible to describe all of mathematics in one (or even several) MSE answer(s). To give you a vague idea, look at Wikipedia's list of areas of mathematics : calculus is really just a small part of one of these areas. – Arnaud D. Aug 10 '18 at 10:46
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    "how much math is there left to learn beyond calculus?" Most of it. Just take a look on the several tags of this site to give a glance at it. – Javi Aug 10 '18 at 10:49
  • Mathematics is way beyond calculus, even if we interpret calculus here to mean not merely calculus on $\mathbf R,$ which is what you already know. We can do calculus in sufficiently nice, more general, spaces, not just on the real line -- these are called manifolds. Also, even if we throw in differential equations as part of calculus, the remaining mass of mathematical knowledge still exceeds it. As you study more, you will eventually realise that the scope of mathematics is literally unlimited. Also, modern mathematics is really something along the lines of what analysis is to calculus – Allawonder Aug 10 '18 at 11:02
  • (continued from above) -- it's not the calculations so much as is the structure of systems of mathematical objects that catches ones fancy or that one has a mind in applying to the study of some other structure. – Allawonder Aug 10 '18 at 11:05

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Disclaimer- I am not familiar with the content of the courses mentioned, but the answer should stand by itself.

For the sake of clarity, I shall answer your question in two parts- 1. How much math is there beyond differential equations ?

Simply put, more than anyone can ever hope to learn in a lifetime. Even ignoring all other branches of Mathematics, there is way more to Calculus, then you can possibly be taught in the mentioned courses - partial differential equations, vector calculus, rigorous foundations using set theory and sequences, various definitions and types of integrability and so on. Of course all that is before you encounter other branches of math like linear algebra, linear programming, advanced statistics, Kolmogorov probability ... and of course, abstract algebra. There are several advanced topics in each of these fields, and you can find our more about them at mathstackexxhange here, mit ocw and a ton of other places. Mathematicians like Archimedes, Aryabhat,Newton ,Leibnitz,Euler ,Ramanujan, etc spent years studying and developing those.

Tip- Try googling 'Math'.( Also check out the wikipedia page) ( No offence, but you need to do that)

  1. How hard is it? That depends on where you stop learning. Everything after that would be impossibly hard, everything before it will be child's play. On a more serious note, you will find trouble once you reach the rigorous theories and abstract algebra taught starting from college courses.

PS- Just don't give up :D.

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