Which parts of Calcs I-III are most important to remember for higher level math?

[raluu]

Hi, I realize that this might be a silly question to ask without some proper background so:
I'm a rising sophomore who is reviewing Calculus I-III right now. I first took Calculus as a junior in high school under the AP Calculus BC curriculum. I then took multivariate calculus and a semester of differential equations in the following senior year.

The problem is, I've stopped retaining everything I learn in math and I was never really that solid on calculus to begin with. It's been a while so my memory of everything was shaky until I my brief review the past few weeks. I've been using Stewart's because that was the same textbook I used in high school. I plan on majoring in math but I'm not quite sure which branch of math I would like to go into. My school seems limited in math classes anyway and we don't have an applied math major; it's just a BA in math and that's it.

So in my review I have been skipping over some parts of the textbook I imagined wasn't ever going to show up again. I mostly want to focus on the parts that I need to be solid on for the future. So if math majors or anyone with experience in higher level math can tell me what in Calc I-III I really need to be know like the back of the hand, I would greatly appreciate it.

 

[chiro]

Hey raluu and welcome to the forums.

The most important parts IMO are the concepts rather than the identities. Knowing what the differentials correspond to and they relate to particular identities is important.

All the applications of calculus like areas, volumes, arc-lengths and so on are based on taking the appropriate limits, or summations and expressing these as integrals.

With the multi-variable identities, it's important to know the context of the linear operator of derivatives and what this means in the context of linear algebra. If you have a good enough understanding of linear algebra, this will be enough to put the differentiation into context.

The other thing is to understand geometry with respect to cross products, inner products, projections in 3D geometry and how limits can be applied to this concepts.

Again, just like the single variable calculus, understanding what the limits correspond to will set you up if you want to nut out the proofs or identities on your own and it will make sense if this premise is down first.

There is a book called "Div, Grad, Curl and all that" that is recommended for more intuition of vector calculus, that you'd probably benefit from.

 

[micromass]

Instead of going throught Stewart, why not go to Spivak?? It's a book that every math major should definitely know about. And maybe you'll find Spivak much more enjoyable than Stewart.

 

[raluu]

Thanks for the advice chiro and micromass.

Ah gosh, so limits are essential? I have a good grasp of how to do them but I never really understood their importance or application in areas of math or science. Stuff like this worries me because I can do the practice problems but I feel like I'm missing something.

I would go through Spivak also but I'm only 4 chapters until completion of Stewart, so it seems a bit too redundant at this stage. I might find Spivak and use that for review on the parts of Calculus I struggle with while reading through Stewart(Series in particular *sigh*)

 

[jgens]

I disagree that reading Stewart makes reading Spivak redundant. The way Stewart presents calculus is almost useless to future mathematicians and his book emphasizes the computational aspects of the subject instead of the theory. On the other hand, the book by Spivak is a much more honest account of what real mathematics is like and his book has the tendency to emphasize the important material in calculus.

Anyway, as far as the multivariable stuff goes, you should definitely know things like the change of variables formula for integration and also things like the inverse function theorem. These are some useful tools to know in differential geometry and topology.

 

[genericusrnme]

Ah gosh, so limits are essential? I have a good grasp of how to do them but I never really understood their importance or application in areas of math or science. Stuff like this worries me because I can do the practice problems but I feel like I'm missing something.

I would go through Spivak also but I'm only 4 chapters until completion of Stewart, so it seems a bit too redundant at this stage. I might find Spivak and use that for review on the parts of Calculus I struggle with while reading through Stewart(Series in particular *sigh*)

Calculus is pretty much the study of limits, both derivatives and integrals are limits of one kind or another

iirc steward doesn't really contain any analysisy rigor, not like spivak does and analysis is really where you're going to end up if you plan on studying maths at a higher level so I'd say spivak > stewart in that respect

 

[chiro]

Ah gosh, so limits are essential? I have a good grasp of how to do them but I never really understood their importance or application in areas of math or science. Stuff like this worries me because I can do the practice problems but I feel like I'm missing something.

The thing about all the results in calculus is that we look at some model and then look at what is actually changing with respect to whatever and use the framework of calculus to calculate the measure in question.

As an example, let's look at the arc-length formula. This formula in a normal cartesian geometry (think of all the axis at right angles like you have in an (x,y) graph) is based on using pythagoras' theorem and then making the length of the hypotenuse be in correspondence with a limit.

So let's say we we calculate s^2 = x^2 + y^2 for some x and y being the length in some interval of a curve. What we do is we let s tend to zero and then use calculus to get the arc-length in terms of the differentials and eventually in terms of derivatives which we can calculate using all of those results worked out before.

The important thing is what the limit means and what it relates to. In the above instance it related to finding the sum of all parts of the lengths of the curve as we shrink the interval of interest down to zero and then we add up all of these infinitesimal s values where ds = SQRT(dx^2 + dy^2).