Keys for Using Maths Effectively

[jmason52]

This may be a dumb question. If so, I apologize.

I have now completed all the basic math courses, calculus thru DE, and am taking upper division physics courses. All my math was application based, ie solving problems; we did no proofs whatsoever. So now, while I am able to solve directed problems, when I look at a complex problem, I am sometimes at a loss as to how to start the calculations.

Thus my question: are there some keys to look for that would clue one in that this problem might be solved via integration, or this one using a differential equation, etc? Or is this just an indicator that maybe I didnt learn the original information as well as I should have?

Thanks for any insights.

 

[chiro]

Hey jmason52 and welcome to the forums.

In terms of taking an expression and solving it, a lot of this is just mathematics which involves transformations of many kinds and manipulation of language (i.e. the symbols).

As for taking a physical problem and converting it into mathematics, the best thing would be to identify exactly what is changing with respect to what else and how its changing: this information will give you the model along with any initial or boundary conditions.

A lot of the applications of calculus in first year like arc-length and volumes are just this: you identify what is changing with respect to what and then take limits to get the expression in terms of a calculus one (differential or integral).

 

[homeomorphic]

So now, while I am able to solve directed problems, when I look at a complex problem, I am sometimes at a loss as to how to start the calculations.

You're starting to experience what serious thinking is like. When you do real thinking, you may be stuck for days or even weeks, but eventually, you figure it out after mulling over it and taking breaks to do other things. That's just the way it is, to some extent. It's not calculus where you can just do the integral in 10 minutes if you know what you're doing without getting stuck or maybe just having to think for a moment here and there and come up with the answer.

Thus my question: are there some keys to look for that would clue one in that this problem might be solved via integration, or this one using a differential equation, etc?

I would say so. But I don't know if it's really "keys" or if it's just something you learn by experience, like riding a bike or playing an instrument. Integration is like adding. When you do a Riemann sum, you add up a bunch of areas. A lot of times, you can think about it by taking a discrete approximation and then taking a limit. So, if you had a bunch or electric charges, sitting at discrete points, to find the electric field, you would add up the electric field due to each charge. But if the charge distribution is continuous, you have to take a limit and it goes from a sum to an integral. The way to get a feel for these things is probably to just do a thousand examples and by then, the thousand and first example is just obvious.

Or is this just an indicator that maybe I didnt learn the original information as well as I should have?

It's possible, but hard to tell without more information.