This might be difficult for me to put into words, but bear with me because I think it's an important question.

Among the many people who study math, I am one of them. I'm not particularly advanced but I enjoy it. I'm working through calculus 2 in my free time. I do it for fun.

It occurred to me: I have a certain amount of ideas in my head: calculus, trig, geometry. These are all tools that required hundreds of years of development. I've developed my ability to reason as best as I can so far.

However, I find that I'm not capable of very much, other than I can work through a textbook, I can solve the problems, maybe even sometimes in clever ways that surprise me. But what can I do? If I am capable of more, I don't know it.

The question, more specifically, is this: if someone has studied up to, say, calc 2, what should they be capable of?

Mathematicians and physicists over the years have achieved much more with much less. So what I'm saying is, with what I know, I feel like I should be capable of more and I'm sure there must be students out there that feel the same.

I'd like to think that with knowledge comes more ability than just regurgitating their studies or passing contrived tests. Insight, intuition should lead them to discover new-found abilities.

Well I hope I've made my question clear and, if I have not, feel free to delete it. Hopefully if I've left any gaps, then the reader can read between the lines and understand what I'm getting at.

Okay think of it this way. Let's say someone has a powerful computer and they're using it as a paperweight. How could they use this resource to the fullest?

I'm expecting question of a format like, "well, if you have trig under your belt, I'd expect you to be able to give me a good estimate of the size of the Earth. You have the tools for it."

Thanks for reading, it's a bit long-winded.


closed as not constructive by Austin Mohr, anon, lhf, Asaf Karagila, Zev Chonoles Feb 15 '12 at 1:37

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    $\begingroup$ I feel this is a bit too soft a question, but: what you can do is really up to you. You encounter some problem/application that interests you in the course of reading different things, and you see if you have the tools necessary for tackling. If you don't, you prepare; if you do, then you can start experimenting. With luck, your experiments might turn out to be something special. $\endgroup$ – J. M. is a poor mathematician Feb 14 '12 at 5:42
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    $\begingroup$ One unexpected and very important side effect of having some background in more advanced mathematics is that you get true mastery of the elementary stuff. Many real world problems require only complete control of very basic mathematics. But not all that many people, even among those who "did well" in high school mathematics but did not go on, have enough control over that material to be able to solve real problems with confidence. $\endgroup$ – André Nicolas Feb 14 '12 at 5:54
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    $\begingroup$ What can you do with your math knowledge? You mean like doing even more math, or real-world jobs, or taking over the world, or what? This is way too broad in my opinion. $\endgroup$ – anon Feb 14 '12 at 6:00
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    $\begingroup$ I don’t think that it’s a bad question to ask, but it’s almost impossible to answer. Ideally you should be able to spot problems that can be solved by the tools that you’ve studied, but in practice @André’s observation is absolutely correct. Typically you acquire real mastery of a set of tools only when you start using them routinely in some more advanced context. $\endgroup$ – Brian M. Scott Feb 14 '12 at 6:50
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    $\begingroup$ One thing that's fun to do is write your own physics simulations. $\endgroup$ – Rahul Feb 14 '12 at 8:14

In my opinion, Calculus (SV, MV, Diff. Eqns) and Linear Algebra forms the basis of most things in Engineering (I mean basic engineering. Not high end stuff, you can get there later though) However, mathematics acts as hygiene and not a motivator. Let me explain what I mean:

Pick up any field of engineering. Let us assume you chose Mechanical Engineering. The major components of ME are Heat Transfer/Fluid Mechanics and maybe newer topics like Robotics, CAD CAM or whatever. Among all of these, the aforementioned mathematical topics suffice to understand all that is being said. But effort is required to link an equation with the "real world".

In you multivariable/differential equation course, you probably learnt how to solve $ x\frac{dy}{dx} = 0$. You may be answers but they won't have any physical intuition. This same equation might show up in heat transfer where $y$ would signify the temperature at a given point. If you know math, you are 1 good book away from knowing the basics of an engineering field. The absence of math knowledge (no matter how good your engineering intuition is) is going to cause problems in the long run.

To cite another example: Say you pick up "rocket science", the prerequisites for basic astrodynamics is Calculus with a little physics (Thermodynamics). There will be people who point out that Calculus and LA are not enough and that one needs other subjects to succeed in Engineering. This is not entirely true. You might need a few basic concepts from Probability/Statistics/Topology etc. but they can be learnt on an ad-hoc basic.

Note: While the theme of SE is not to spurt out opinions but facts, this question invited one so I provided.


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