What to focus on in calculus (as a physics major)?

[Ascendant78]

I just finished up calc II this semester. So far, seems fairly easy and I got an A no problem. However, there was a lot of material in each of the chapters that we didn't cover as well as more difficult problems in the sections that we didn't work on. Our professor told us it really isn't feasible to try to learn how to do the most difficult sections of most chapters because it is just too time-consuming and that any of the more in-depth material that you need to know for your major will be covered more extensively later on.

So, on that note, I do want to put at least some of my study time into calc, but I also don't want to waste time on stuff I won't use much later on. I want to start learning computer programming over the break and start working on the Morin book for classical mechanics. So, what type of stuff from calc I to calc II is really important later on? What would be worthwhile putting practice into to improve on? I have heard so far that the hyperbolics really aren't too important since you hardly ever see them later on, but I'm not sure what I should try to improve on?

I know this is difficult to answer generally speaking, but I guess if I could get an idea of what I will see more of later on in physics, I will know where I could most productively spend my time on calc? The only feedback beyond the hyperbolics that I have gotten so far is that natural log, exponential functions, and sin/cos functions pop up a lot in physics, so it's good to have the derivatives and integrals of those down solid. Any additional feedback would be greatly appreciated.

 

[jbrussell93]

I would agree that hyperbolics will pop up a little bit later on. For example you can write the solution to the critically damped harmonic oscillator in terms of hyperbolics. Additionally, in classical mechanics I recall running into some integrals that required hyperbolic trig substitutions.

As far as important things you'll use from calc 1 and 2... pretty much everything you've learned about differentiating and integrating obviously. You'll definitely use the different integration techniques you've learned. A solid understanding of exponential and natural log become very important as well.

Honestly, the meat and potatoes of the math for a physics degree comes from vector calculus (calc 3) and differential equations. Those were the two most useful math classes for my upper division physics classes but you really need the solid foundation from calc 1 and 2. I would also suggest taking a semester or two of math methods from the physics department if your school offers them.

 

[Sentin3l]

Honestly know everything from calc I and II. They are the fundamentals for the rest of your mathematical career and I can promise you will use all of it at one point or another.

 

[Ascendant78]

Well thank you both. As far as knowing everything, I already have the trig identities/formulas down along with all the various methods we used like logarithmic differentiation, integration by parts, trig substitution, etc. I just know that practicing it can show me some little tricks here or there that I may not have seen yet, or see problems that require the application of things I haven't done yet. On the other hand, I have so many other things I want to study and only so many hours to study it, I don't want to waste time over-studying calc when I could put it to better use learning computer programming, physics, etc.

I also like what you (jbrussell) said about calc 3 and differential equations, as I will be taking both during this next semester along with physics 2. I'm looking forward to seeing more math that will be applicable to the physics I'm learning. Again, I thank you two for the info and I'm thinking I will probably just skim over my notes for the break rather than spend too much time going over actual problems.

 

[Student100]

I remember bits and pieces of the applications of derivatives, applications of integrals, and a lot of trig/basic vectors/algebra/coordinate manipulation in the first course of physics. Mostly I remember there was tons of trig.

It sounds like you're good in the math department to take physics, you'd be better off practicing solving physical problems, with maybe some related rates stuff, max min problems, work, force/fluid pressure problems. Learn to derive all the trig indents, and practice some problems from your precalculus book on vectors/law of sines/cosines/basic trig manipulation.

That should put you on solid footing.

As far as what your professor said, that sounds legit to me. Calculus is a huge subject, it's impossible to cover everything in depth. So when you get time it's good to work through the books again, and try to solidify your knowledge base as much as possible. I've been done with calc three/vector/de for a while and I still go through all the books learning or relearning things I haven't used in a while. Lately I've been fixated on vectors and tensors though. Right now in my opinion it'd be better for you to try to make your entrance into the world of physics as smooth as possible by doing the above.

 

[TomServo]

It's easier to list what not to focus on, and those would be the integrals of volumes you do before multivariable. Pipe method and that.

The multivariable methods are easier.

 

[WannabeNewton]

Easily the most important things you should know like the back of your hand (both in terms of theory and computation) are Taylor series. They are one of the most important tools in physics.

 

[Student100]

It's easier to list what not to focus on, and those would be the integrals of volumes you do before multivariable. Pipe method and that.

The multivariable methods are easier.

Yeah but he hasn't done multivariable yet. :tongue:

 

[jbrussell93]

I also like what you (jbrussell) said about calc 3 and differential equations, as I will be taking both during this next semester along with physics 2. I'm looking forward to seeing more math that will be applicable to the physics I'm learning.

That will be a great semester. I took calc 3 and physics 2 together and that's what made me switch my major from engineering to physics.

Also I think focusing on programming is a great idea. I taught myself some programming one summer and it has been very useful.

 

[Student100]

Easily the most important things you should know like the back of your hand (both in terms of theory and computation) are Taylor series. They are one of the most important tools in physics.

I agree with WBN. I've noticed that most of my HW problems would be a MacLaurin series, but on the tests you need Taylor's. Haha physics professors are tricky like that.

 

[Jorriss]

As WBN noted, series in general are very important as well as the geometrical interpretations of derivatives and integrals. That being said, calculus is basic and all of it should be understood relatively cold.

 

[Ascendant78]

Well thanks for all the information here. It has been extremely insightful!

As far as Taylor and MacLaurin series, those don't sound familiar at all to me. Not sure if we covered it but he just didn't cover the name with us, but at a quick glance from a web search, it looks pretty foreign to me. The closest of anything we covered that looked like that type of stuff was improper integrals, but we only skimmed over that briefly for about a half an hour last week and then reviewed for the final. I have some MIT opencourseware with it that I will be sure to take a look at. Again, I really appreciate all the feedback from all of you.1

 

[jbrussell93]

Series are typically covered at the end of calc 2. The ideas and applications of series are extremely important. I've never heard of a calc 2 class not covering them. What book did you use? How many semesters of calculus are in your sequence?

 

[Ascendant78]

Series are typically covered at the end of calc 2. The ideas and applications of series are extremely important. I've never heard of a calc 2 class not covering them. What book did you use? How many semesters of calculus are in your sequence?

We used Calculus by Briggs and Cochran. Our sequence is 3, calc I, II, and III. I'm actually watching the lectures from MIT on infinite series. He did cover this stuff briefly, but definitely not nearly as in-depth as this video has. Still haven't gotten to the Taylor series part, but it is listed later on in the lecture. It doesn't seem to have the MacLaurin though.

 

[Student100]

MacLaurin series is just a special case of the Taylor series, you'll see when you get to Taylor's series in the videos I hope. It's normal for it to be introduced as an introduction to the Taylor series.

 

[bp_psy]

Trig,complex exponential manipulations,integration by parts and everything about series.

 

[TomServo]

Ditto Taylor series.

I really need a review in that and diffy qs before grad school...

 

[Ascendant78]

Well, I learned about infinite series and Taylor series from an MIT single-variable course. However, they only skimmed over it themselves in the last couple lectures and didn't really do much of anything as far as utilizing it, showing its purposes, etc. Is that pretty much all I need at this point is an awareness of those things so that when I see them later on, I will know what I am looking at?

 

[Student100]

Watch this too, might help you some.

 

[Ascendant78]

Watch this too, might help you some.

Thanks a lot for the link. I watched it and the additional things I took from it beyond what the MIT one gave me is that Taylor Series is (in some cases) only an approximation, doesn't work for some functions (asymptotes can cause problems for example), but that overall, it gives you a different way of viewing a function through a series.

Since everyone seems to be emphasizing the importance of it, I think I'm going to try to find where it's covered in our calc book and work on some problems involving it. I'll also see if that MIT opencourseware has any problems to work on.

 

[WannabeNewton]

If you go through Morin, as you said at the beginning of the thread, then you'll be using Taylor approximations over and over and over and over even if you don't recognize it immediately. The most prominent forms of Taylor approximations that will show up in classical mechanics books like Morin and Kleppner are binomial approximations and small angle approximations.

As far physics texts go (at least at this level) you'll basically be taking for granted that the functions you Taylor expand are analytic.