Where to learn higher calculus?

glenn101 · 2008-08-01 12:44:32

Hey guys,
So far in my maths class we have finished up calculus for the year, but I want to continue to learn.
All we covered was;
Limits,
Finding the derivative by first principles,
Finding the derivative by rule,
and antidifferentiation (indefinate integrals).

Any ideas of where to study higher from where I am?

ZHero · 2008-08-01 15:05:11

You seem to be an Ambitious Aspirant!
Have you done your +2? If yes, then go for Engineering!
That's the only way 'I' know...!
Hope there are some websites that may help you...

Ricky · 2008-08-02 08:13:03

Any calculus book should contain more than that. Most cover applications of integrals (motion, volume, arc length), improper integrals, and an introduction to infinite sequences/series.

I don't know how your class is structured, but in an introductory calculus class, you typically don't cover "hard" integration. Do you know:

Integration by parts
Trigonometric substitution
Partial Fractions
Integration by tables

Those are pretty easy to learn once you get integration down. The other place to go (if you've done this) is different types of calculus. This list is ordered by difficulty (and some of the higher ones require lower as a prereq)

Multivariable Calculus
Differential equations
Vector Calculus

Now if you get really up there, you can go on to real analysis. As part of this, you can study a wide variety of topics:

Lebesgue measure/integrals
Riemann-Stieltjes integrals
Differential forms
Smooth manifolds
Partial differential equations

But this way down the road for you, and would only be studied if you choose to become a mathematician or certain type of engineer.

glenn101 · 2008-08-02 12:11:11

Well I understand what your saying Ricky,
in my maths class our calculus section was small.

These are the chapters we did:
Rates of change first (without differentiation)
Differentiation (first principles, derivatives by rule, antidifferentiation) *no integration*
and then applications of differentiation ie stationary points, maximums and minimums.

Its not really a calulus oriented textbook I mean we have done
Quadratics, Polynomials, Cubics, quartics, probability and our next topic is circular functions.
So we only really covered the basics, but I was intrigued to learn more after it.

All this is done in my maths methods class by the way, and I do 2 different maths classes. Here in Australia we have 3 different levels of maths in year 11,12 there is;
starting from easiest to hardest:
maths further,
maths methods,
maths specialist.

I do methods and specialist, but my specialist class is in someway unstructured it doesn't seem like we are doing those hard aspects, we have done;

Matrices, Simultaneous Triples, Partial Fractions, Transformations, and now logarithms and difference equations. In specialist we havn't done any calculus as such, but I hear next year we do integration and differential equations etc.

So yeah, I'm just interested to learn more calculus, as when I finish I am going to enter an engineering stream, most definatly electronic engineering like Zhero:)

Last edited by glenn101 (2008-08-02 12:14:04)

Identity · 2008-08-02 14:23:33

So I'm guessing you're in year 11 at the moment? Next year I strongly recommend you try to get into Melb Uni Extension Maths or Monash Uni Extension Maths. Unfortunately they have taken a lot of calculus out of the course since the past few years, but it still has advanced calculus and other fun: complex numbers, matrices etc.

Anyway, near the end of the year you will cover basic integration in Methods. In Methods 3/4 you will cover pretty much every basic integral but you won't learn advanced techniques such as substitution or integration by parts.
In Specialist 3/4 you will cover substitution but not integration by parts ( i think). You will also cover vector calculus.

Here is a forum discussing the university extension courses. It's also a general VCE forum. You may need to register though.
http://vcenotes.com/forum/index.php/board,96.0.html

glenn101 · 2008-08-04 22:07:25

Thanks identity for your reply:)
yes I am in year 11 at the moment, but how do I apply for uni maths next year? does it become one of my 5 subjects and replace another? how does it work?

I really enjoy maths methods and maths specialist, I want to go into electronic engineering in uni as I like electronics and the engineering part goes into advanced maths. So would this uni maths subject be good?

Last edited by glenn101 (2008-08-04 22:10:16)

Identity · 2008-08-05 00:51:30

An extension subject automatically becomes your 6th subject in VCE, the maximum aggregate score you can get from it is 5.5. This may not seem much but if you had another normal VCE subject in your 5th or 6th spots then the max you can get most of the time is 10% of 50 = 5. The extra 10% increments by your 5th and 6th subjects can be crucial in pushing your ENTER 99+.

http://www.vcaa.vic.edu.au/vce/studies/ … nsion.html

Melbourne, Monash, and Victoria Universities offer the courses. You need to go to the following sites and read the information. Talk to your teacher about it as well to see if it's the best idea.
e.g
http://www.monash.edu.au/study/enhancem … ation.html
http://www.services.unimelb.edu.au/umep … ation.html
[Um, I don't know why the VU link is broken, but Monash and Melbourne are more prestigious anyway]

Also on offer is Extension Physics. Perhaps if you're doing Electronic Engineering this might be more useful to you than Extension Maths, but it's your choice. Having said that, any extension study would be a great head-start into that field.
All Extension Courses cover a 1st Year University course in that particular subject. Another fantastic benefit of doing a Uni Extension is that it actually gives you Credit for if you continue to study a degree in that subject at the university! So you will reduce both the time and fees required in university to fund those 1st year Credits.
If a 1st Year Uni course sounds like a lot of hard work and intensity, don't worry. I've spoken to people on vcenotes.com and all of them say it requires the same or less work than a VCE subject. Especially Monash Maths

Oh BTW, I also heard that Melbourne University Engineering is really bad. It is very theoretical and supposedly people have been demotivated by it. So if you choose to do engineering I would recommend Monash.

Last edited by Identity (2008-08-05 01:36:06)

glenn101 · 2008-08-07 19:16:05

Thanks again for the reply Identity:)
Thanks for putting forward your incite into Melbourne uni's engineering, I was considering going there but from what you said, Monash sounds to be the better choice, so I will see what Monash has to offer, so thanks for that.:D

Oh and what is La Trobe uni like? they also have engineering as well as Deakin.

Last edited by glenn101 (2008-08-07 19:19:44)

mikau · 2008-08-21 06:17:17

I thought Vector Calculus WAS multivariable calculus!

Whats the difference?

Ricky · 2008-08-21 07:41:45

Multivariable calculus is calculus in 2 or 3 (sometimes, though rarely 4) dimensions. Vector calculus is n-dimensional calculus. Some things you don't learn about in multivariable calculus:

Curl
Div
Paths in space
C1 and smoothness
Integrating around a surface.
Green, Stokes, and Gauss's theorems.
Jacobian and change of coordinates (in the general setting)

mikau · 2008-08-21 12:21:23

Okay, but does that mean vectors in R^n/ C^n, or does it deal with the elements of any vector space?

Seems to me you should be able to develop a calculus for any vector space over a Field that contains an infinite number of elements between a and b...

Ricky · 2008-08-21 13:30:54

Seems to me you should be able to develop a calculus for any vector space over a Field that contains an infinite number of elements between a and b...

First off, you need to be able to define what a function is. Specifically, you need to be able to define continuous functions. Integration is impossible on a function that is not continuous on a vast majority of points, let alone on a function that you can't say whether or not it is continuous. This means that you need at the very least, a topology on the field.

Unfortunately the other requirements require knowledge of topology and analysis. I just can't seem to boil them down to make sense to someone who hasn't studied these. But I'll just list them, and maybe I'll find a way later. The topology must be:

Hausdorff.
Second countable.
Diffeomorphic to the reals.

Math Is Fun Forum

#1 2008-08-01 12:44:32

Where to learn higher calculus?

#2 2008-08-01 15:05:11

Re: Where to learn higher calculus?

#3 2008-08-02 08:13:03

Re: Where to learn higher calculus?

#4 2008-08-02 12:11:11

Re: Where to learn higher calculus?

#5 2008-08-02 14:23:33

Re: Where to learn higher calculus?

#6 2008-08-04 22:07:25

Re: Where to learn higher calculus?

#7 2008-08-05 00:51:30

Re: Where to learn higher calculus?

#8 2008-08-07 19:16:05

Re: Where to learn higher calculus?

#9 2008-08-21 06:17:17

Re: Where to learn higher calculus?

#10 2008-08-21 07:41:45

Re: Where to learn higher calculus?

#11 2008-08-21 12:21:23

Re: Where to learn higher calculus?

#12 2008-08-21 13:30:54

Re: Where to learn higher calculus?

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