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how do you overcome the difficulty in learning it?

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- Thread starter cks
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- #1

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how do you overcome the difficulty in learning it?

- #2

HallsofIvy

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- #3

Gib Z

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And for everything, try to understand them in several ways. Thats a tip i read from Terry Tao's blog. Ie For all theorems and definitions, know what they mean algebraically, geometrically, intuitively etc etc. EG The derivative geometrically is the slope of the tangent at a point. Algebraically is it the limit [tex]\lim_{h\to 0} \frac{ f(x+h) - f(x)}{h}[/tex]. Intuitively it is the instantaneous rate of change of a function of one variable with respect to another.

Or the mean value theorem for integrals: Geometrically it is the area of the "mean value" rectangle. Algebraically it is [tex] \frac{1}{b-a} \int^b_a f(x) dx[/tex]. Intuitively it is the average value of the function between the bounds b and a.

You get the point I'm sure.

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So what other advice can i get from u guys?(and gals)

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Gib Z

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- #6

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but i havent done euler's formula yet :S

maybe later. uni started only 2weeks back

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Gib Z

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- #8

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yeah thx a lot, if i get any problem understanding i'll aks :)

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- #10

mathwonk

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get a more explanatory book, like one by george simmons.

- #11

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Don't let it get too abstract. For each definition/theorem come up with the simplest, (but non-trivial) concrete example, and make sure you understand why the definition/theorem makes sense.

Then try to come up with the most likely counterexamples to the theorem (or definition), and convince yourself that those counterexamples fail.

I read thru Munkres' Topology before Rudin's Priciples of Mathematical Analysis. Munkres does have plenty of good examples, and makes a good companion text.

Then try to come up with the most likely counterexamples to the theorem (or definition), and convince yourself that those counterexamples fail.

I read thru Munkres' Topology before Rudin's Priciples of Mathematical Analysis. Munkres does have plenty of good examples, and makes a good companion text.

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- #12

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And for everything, try to understand them in several ways. Thats a tip i read from Terry Tao's blog. Ie For all theorems and definitions, know what they mean algebraically, geometrically, intuitively etc etc. EG The derivative geometrically is the slope of the tangent at a point. Algebraically is it the limit [tex]\lim_{h\to 0} \frac{ f(x+h) - f(x)}{h}[/tex]. Intuitively it is the instantaneous rate of change of a function of one variable with respect to another.

Or the mean value theorem for integrals: Geometrically it is the area of the "mean value" rectangle. Algebraically it is [tex] \frac{1}{b-a} \int^b_a f(x) dx[/tex]. Intuitively it is the average value of the function between the bounds b and a.

You get the point I'm sure.

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- #14

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get a more explanatory book, like one by george simmons.

I hvae borrowed a book of his. there are a lot of explanations in the first few chapters, and clarify a lot of doubts. thanks

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The title of the book by George F. Simmons is

Introduction to Topology and Modern Analysis

Introduction to Topology and Modern Analysis

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I major in physics but am taking this course because I believe it can provides me a solid foundation for my future research in Quantum Mechanics.

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Ok will look for the book u guys recommended

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I major in physics but am taking this course because I believe it can provides me a solid foundation for my future research in Quantum Mechanics.

You should take an introductory course in analysis. Frankly, the real analysis course is going to be very demanding and will expect you to be at a certain level of mathematical competency.

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- #20

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I second the Pugh recommendation. It covers intro real analysis in a clearer way than Rudin.

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- #21

Haelfix

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The hard part is both the great strength and great weakness of Rudin. Just using his text, you are forced to memorize and learn the gritty details of the proofs ad nauseum, but at the same time you can fall a bit into 'rigor' mortis and you forget about the intuition.

That was the point roughly in my career where I decided I wanted to be a physicist and not a mathematician. I stuck with it for a few more years, but that was the turning point I think.

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Thank you to all you guys, I will put all my efforts in it.

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I often see this formula, [tex]e^{ix} = \cos x + i \sin x[/tex]

but I forgot its meaning.Can sb tell me is it

only a definition or can be proven to be so?

- #24

Gib Z

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That formula can be proven by letting ix be the variable in the maclaurin series for e^x, and showing that it is the same as the maclaurin series for cos x + i sin x.

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