Difference between integral calc and multivariable calc?

[JoeyCentral]

Hello, I am taking integral calculus at the moment at my college, and I was just wondering what is the difference between integral calculus and multivariable calculus?

 

[axmls]

Typically, "integral calculus" refers to a course on integral calculus in one variable, usually consisting of techniques of integration, applications, and infinite sequences and series. Multivariable calculus courses typically cover both differential and integral calculus in more than one variable, with a particular focus on functions of two and three variables (for instance, single variable would be $y = f(x)$ whereas multivariable may be $z = f(x, y)$).

 

[JoeyCentral]

I see. How much of integral calc do you need to learn multivariable calc? How is solving for a multi-variable problem different from solving a single-variable problem?

 

[axmls]

Everything you use in single variable calculus (most of everything, at least), you will also use in multivariable calculus, and the difference is that multivariable calculus will have more than one variable. That means you must know your derivative rules and you must know your integral rules.

In multivariable differential calculus, for example, you're concerned with partial derivatives. Instead of having to worry about the rate of change in one direction, there are now an infinite amount of directions you can go from one point, so it is necessary to develop methods to measure these changes.

 

[HallsofIvy]

I would not say just "integral" Calculus but "differential and integral" Calculus. In "differential and integral" Calculus you learn to differentiate and integrate functions of a single variable in "multivariable" Calculus, you learn to differentiate and integrate functions of more than one variable. But those, of course, are just extensions of the concepts for single variables.

 

[RJLiberator]

Yes -- The major difference is multiple variables. Usually, universities will focus strongly on the multivariable aspect and assume you have a working knowledge of integration/differentiation once reaching that point.
For many I speak with and from my experience, integral calculus was more difficult then multivariable calculus. Integral calculus is a lot of new topics whereas multivariable is like an extension of them into more dimensions.

 

[JoeyCentral]

Yeah, my math teacher did warn us that Calculus 2 is arguably the hardest Calculus class in the series. I understand why she says that, though so far, I am doing good in my class and I feel like I am understanding everything good. I feel the only time I may have trouble is when we start going into sequences and series. I didn't do bad at the ∑ concept, though I did take a quick glance at the taylor series, and it does look pretty intense.

I am curious to find out how to deal with multi-variable problems and how the steps differs from solving them with single variables. If anybody has the time to spare to show me an example, that would be great.

 

[RJLiberator]

Check out Khan academy --> subjects --> Multivariable calculus --> Double and triple integrals

you will see how integrals are taking within multivariables.

 

[onethatyawns]

Yes -- The major difference is multiple variables. Usually, universities will focus strongly on the multivariable aspect and assume you have a working knowledge of integration/differentiation once reaching that point.
For many I speak with and from my experience, integral calculus was more difficult then multivariable calculus. Integral calculus is a lot of new topics whereas multivariable is like an extension of them into more dimensions.

I agree. I'm learning integral calculus (in Calc 2) right now, and there are a ton of new things. It could be argued that this is only because derivative calculus is not taught properly, in such a way that makes integrals obvious, or maybe integrals are just so abstract (or more accurate, a seemingly endless series of iterating) that it really is a harder concept to master.

 

[foobar99]

It is a bit of a didactic distinction without foundation, maybe tied to history.

Typically, intro calc is divided into differential calc then integral calc. Differential involves only derivatives, built up on the $\lim \frac{f(x+h)-f(x)}{h}$ as h goes to zero. this is slope identification. Integral calc is finding the area under a curve using the limiting sum of rectangles. It is then argued that, by some miracle, the integral is functionally the same as the anti-derivative from differential calc. This gives us a quick way to avoid laying out rectangles and summing, since we can, by rules and insight (much like factoring in algebra) figure out how to quickly get the anti-derivative of a function we wish to integrate. This is all calculus 1.

Multivariate calc is a lot more complex and subtle. Usually in calc 1 there are only two dimensions, X and Y, and y = f(x). Mathematicians would more accurately say this one dimensional calculus, where functions map from $R^1 \rightarrow R^1$ - the real line to the real line. In multivariate you now have an n-dimensional domain mapping to another space, typically (but not always!) on dimensional.

The level of abstraction is quite a bit higher. In addition to all this space mapping, one ends up dealing with new 'things', like vectors and partial derivatives. On top of that there are things like changes of coordinates. For example, a lot of work goes into mapping from Euclidean (rectangular) to polar and cylindrical coordinates. One finds that these mappings themselves are not volume-preserving, so one needs to account for a whole bunch of structure and subtlety that never happens in single varialbe (search 'Jacobian'). In calc 1, the only coordinate change one might run into is a trivial one like $x' = x + a$, which really means only changing the limits of integration. But, move from rectangular to polar and a whole lot more changes...

If I had to describe the pieces, I would do it as such:

1) Calc 1 - diff and integral single variable
2) multivariate
3) complex calculus (this gets really, really weird)
4) measure-theoretic (Riemann-Stieltjes) where one now no longer deals with unit differentials (so you can integrate w.r.t. things like dx^2)

 

[HallsofIvy]

There are textbooks that start with integral calculus then differential calculus. In some ways the theory of Calculus makes more sense that way, but, of course, actually calculating integrals, before you have the notion of an "anti-derivative", which, of course, requires you know the "derivative" is extremely tedious.

In any case, I would think that the "difference between "Integral Calculus" and "Multi-variable Calculus" would be clear from the name. "Multi-variable Calculus" covers both differential and integral Calculus with more than one variable.