Math Books

[doogly]

Dan Rockmore's Stalking the Riemann Hypothesis could be good too. The reviews indicate that it can be pretty frustratingly non-mathy - only one equation! I think these reviewers do not actually know what "popular level" means.

Also worth noting that he is a sweet dude who taught my calc 3 class.

[sciencenerd]

Everyday Calculus: Discovering the Hidden Math All Around Us by Oscar E. Fernandez
Seems like a good book

[AngelAttractor]

Can someone please recommend me a book on the subject of nonlinear systems?

[chi-squared]

I've really enjoyed Infinite Dimensional Analysis by Aliprantis and Border.

It has a reasonably comprehensive coverage (at least, as comprehensive as you can get given the breadth of the subject matter) of topology, measure theory and functional analysis.

If anyone has suggestions of texts with a similar bend, I'd love to hear them.

[Farabor]

I'm trying to learn about Banach Lattices (Riesz spaces which are Banach Spaces). The textbook my professor gave me to work from is Peter Meyer-Nieberg's "Banach Lattices", 1st edition, 1991. I'm finding it extremely dense/hard to follow. (I didn't have any problem with Kreyszig's Introductory Functional Analysis, which is my background). Does anyone have other books to recommend that I might find more accessible?

[chi-squared]

I'm trying to learn about Banach Lattices (Riesz spaces which are Banach Spaces). The textbook my professor gave me to work from is Peter Meyer-Nieberg's "Banach Lattices", 1st edition, 1991. I'm finding it extremely dense/hard to follow. (I didn't have any problem with Kreyszig's Introductory Functional Analysis, which is my background). Does anyone have other books to recommend that I might find more accessible?

The book I mention one post above (Aliprantis & Border) covers Banach Lattices. I think that was also one of Aliprantis' main research areas, so it might be worth a look.

[monkey3]

can anyone recommend a good calculus book ?

[Bane Harper]

If you are new to calculus begin with YouTube videos and Pre‑Calculus For Dummies by Yang Kuang

[monkey3]

I like that book already , i mean i was sort of stuck at trigonometry for sometime now
Not sure where it begins or where it ends .

I somehow need to understand trigonometry a bit more deeply , i mean everything in it , the beginning and the end . i hope there is such a thing in trigonometry

Anyway that book looks nice especially this part



Do you know any other good trigonometry books or trigonometry teaching websites ?

[monkey3]

What do you learn after this if your aim is learning calculus ?

[doogly]

What is motivating your study for this? Any particular application or field, or just a pure math curiosity?

[monkey3]

Do you want to see a list of things i have gathered ? i don't know if i should use this thread to post that list of things ...

Are you interested in that list of things ?

Anyway i am trying to learn calculus

[ucim]

What do you learn after this if your aim is learning calculus ?

I'm not sure what you mean by "learn". You posted a great list of trigonometric identities, which are useful tools in actually doing trigonometry. But it almost seems as if you equate memorizing these tables with "learning trig". For example, missing on this page is the unit circle, which visually conveys what "sin" and "cos" etc. mean.

In any case, trig is not a precursor to calculus. It's just that a lot of the useful applications of calculus require trig anyway, and trig gives you a whole bunch of new functions to play calculus with, so it's good to have those tools under your belt first.

Calculus is first and foremost about rates of change. Things change over time (or distance, or whatever), and at any given (one) point there is a certain rate of change, but you can only "see" the change by comparing two points. So, you consider two points "very" close together and get an approximation (because it's changing!)... the closer together your two points the better the approximation. Calculus is about taking this to eleven.

For that, you need the idea of a limit. As "delta x" (the difference between the two points) goes to zero, the value of the expression you are looking for will get closer and closer to some value; that value is the limit.

Trivial case: y=2x. As x approaches 4, y will approach 8.
Problematic case: y=1/x. As x approaches 0, y increases without bound ("approaches infinity").
Useful illustration: y=x²/x.

As x approaches 0, what happens? The bottom makes the expression want to blow up ("approach infinity"), but the top looks well behaved. You can't just divide both sides by x to simplify, because at x=0 you're dividing by 0. However, you can do that as long as x doesn't equal zero. What happens? We simply get y=x in those cases. As x approaches zero, y also approaches zero. The limit of the expression x²/x as x approaches 0 is zero.
More interesting expressions where the limit isn't obvious are available in a zillion textbooks. But that's the idea of a limit.

We then apply that idea to rates of change, to find (say) the slope of a curve. That curve needs to be expressed first in math, for example,
y=3x²
Slope is rise over run; that's easy if you have a straight line. Just pick two points and do arithmetic; the slope is constant. But with this curve, the slope keeps changing. We want an expression for the slope of this curve at any point. To do this, we still pick two points. One of them will be x (the point at which we want the value of the slope), and the other will be nearby: (x+a), where a is small.

Slope is rise over run. The run is easy... it's the horizontal distance between the two points It's just a.
Algebraically, it's (x+a)-x (the x coordinate of the second point, minus the x coordinate of the first point)
The rise is a little harder; we need to use the formula for our original curve and take the difference of the two resulting values. For the first point it's just
3(x)²
For the second point it's
3(x+a)²
Subtract the two, and the rise is:

rise = 3(x+a)² - 3(x)²
rise = (3x² + 6xa + 3a²) - 3(x)²
rise = 6xa + 3a²

so the slope = rise/run = (6xa + 3a²)/a = 6xa/a + 3a²/a

Now, take the limit as a=>0 (as the distance between the two points we used to calculate the slope approximation vanishes)

We get slope = 6x

The a in the denominator can never be zero, but we can see from the expression that the closer we get to a=0, the closer the equation comes to y=6x.

The slope actually increases (linearly in x). As x moves to the right, the slope gets steeper. (and as it moves to the left of zero, it gets steeper in the other direction). When x is zero, the slope is zero, and the curve is at a minimum (or maximum).

That is calculus in action.

No trig involved. But of course trig functions have slopes, and there's lots of fun to be had with calculus on trig functions.

Jose

[monkey3]

Thanks ucim ,

To be honest , i started this account to improve my math from the very beginning ...

Arithmetic
Algebra
Trigonometry
Differentiation
Integration
Differential equation

Spoiler:

whole numbers , natural numbers , integers , rational numbers , irrational numbers .

factors ( reducible), prime factors (irreducible), greatest common factor (same as greatest common divisor ), least common multiple

Monomial , binomial , trinomial , polynomial

simplify , factoring (factorization )

Factoring polynomials

’To factor’ means to break up into multiples.

Methods of Factoring

Factor by Distributive law method



Factor by grouping



Factor by Splitting



Factor by Very Famous Polynomials






After that trigonometry





Calculus derivatives and limits



Calculus integrals

The sad part is i haven't got a list like that for differential equations , i don't even know what i should be looking for ...

[gmalivuk]

I put the middle of that post in [spoiler] tags to make the page a bit easier to read. Please do that whenever a post is going to be more than a couple of screen-heights to scroll through.

[doogly]

Kline's Calculus: An Intuitive and Physical Approach is pretty tight, combo it up with Spivak if you want a little more high falutin delightfulness.

[monkey3]

Thanks for putting it in the spoiler tag , and thanks for the books suggestions

[CTepp]

Can anyone recommend a book on advanced numerical analysis of differential equations? I am particularly interested in one with reasonably fast techniques for stiff non-periodic PDEs with both a finite and infinite [0,Inf) horizon.