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    Post-grad feeling the tiniest bit lost

    I finished my bachelors a 2-3 years back in mathematics and working a pretty good job in New York in the tech industry, but I am having trouble deciding what to do next! My options are either continue down this route and become a product manager, go more heavily into software engineering...
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    Starting out in pde

    Homework Statement Hey I'm trying to get a sense of this problem, just starting pde class: au_x+bu_y+cu=0 Homework Equations The Attempt at a Solution Dunno what to do with that last term
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    Set theory

    Homework Statement Homework Equations An equivalence relation on a set A, is for a,b,c in A if: a~a a~b => b~a a~b and b~c => a~c The Attempt at a Solution It seems uncomplicated, but I don't know how I would write down a proof. The book I'm using is Topics in Algebra, 1st Edition Herstein
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    Studying What to do about textbooks with solutions

    edit: whoops, the title should read, "what to do about textbooks with no solutions" So I'm trying to learn real analysis (using C.C. Pugh's book) and its going alright until I hit the exercises. There are no solutions! How am I supposed to know I proved something correctly? Is there a website...
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    Riemann Lebesgue Theorem

    Homework Statement We have a corollary that if f(x) is in the set of Riemann Integrable functions and g(x) is continuous, then g(f(x)) is also a riemann integrable function Show that if g(x) is piecewise continuous then this is not true Homework Equations Hint: take f to be a ruler...
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    Analysis problem

    Homework Statement Given x > 0 and n \in N, prove that there is a unique y > 0 s.t. y^n = x exists and is unique Homework Equations Hint is given: consider y = 1. u.b. \{s \in R : s^n < x\} The Attempt at a Solution I'm not used to this style of proof (real analysis I), help would be...
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    Modular arith, number theory problem

    Homework Statement Find the number of roots for the equation x^2+1=0 \mod n \: for \: n = 8,9,10,45 Homework Equations The Attempt at a Solution I have no idea where to start. Could someone help me understand?
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    How hard is number theory?

    I'm taking the class next semester, and I heard that number theory is usually a difficult subject. Is that true? If so, how should I approach it?
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    Reimann Zeta Function at 2

    So we were going over geometric series in my calc class (basic, I know), however I was intrigued by one point that the prof. made during lecture \frac{\pi^2}{6} = \sum^{\infty}_{n=1}\frac{1}{n^2} = \zeta (2) That's amazing (at least to me). Looking for the explanation for this, I found a...
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    The Princeton Guide to Mathematics

    by Timothy Gowers. How is the book? I'm thinking of purchasing it for my undergrad math career, can anyone recommend it?
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    Nested Radicals

    I been trying to solve some nested radicals. I've been able to do: \[\sqrt{1+\sqrt{1+\sqrt{1+\sqrt{1+\cdots}}}}\] Which is pretty cool since it equals to the "Golden Ratio" or \[\frac{\sqrt{5}+1}{2}\] But I can't seem to do the following: \[\sqrt{1+\sqrt{2+\sqrt{3+\sqrt{4+\cdots}}}}\] Using a...
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    Anyone know how we can improve our mental math skills?

    The other day I decided to calculate 1.225 x 36 in my head (this was a calculation in one of my problem sets) and I found that it took me a lot longer than I originally thought. So I was wondering if there was any sort of strategy to gradually improve this skill. Here's a strategy I am...
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    Picking a chain from one end

    Homework Statement You have a chain of length 10m with 80kg, how much work does it take to lift this chain from one end to 6ft? Homework Equations \delta = \frac{10}{80} = .125 The Attempt at a Solution W = \int{F(x)}\,dx = \int^{6}_{0}{\delta lg}\,dl = \frac{\delta l^2g}{2}...
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    Mathematica Mathematica Plotting interms of Y

    I'm trying to plot a function in terms of Y, but it always treats y as if it were x. Is there anyway I could "Turn" the graph 90 degrees? Thanks
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    Integral problem

    Homework Statement Evaluate $\displaystyle\int^{1}_{0}{\sqrt{x^2+1}}$ Homework Equations The Attempt at a Solution By trigonometric substitution: $x = \tan{\theta} \rightarrow dx = \sec^2{\theta}\,d\theta$ \[\int^{\frac{\pi}{4}}_{0}{\sec^2{x}\sqrt{\tan^2{\theta}+1}}\,d\theta =...
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    Integral help

    Homework Statement It looks deceptively easy, but I can't seem to get it... \int^{\frac{\pi}{4}}_{0}{\\sec^4{x}\tan^4{x}}\,dx Homework Equations \sec^2{x} = \tan^2{x} +1 The Attempt at a Solution I've tried, but they all end up as trigonometric polynomials
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    Apostol's Calculus

    I've already self-studied calculus, so I wanted to go on and learn it in multiple variables. Generally, Apostol seems to have had good reviews for 'pure, prose learning', but how is it to actually learn from it? And if I do should I read the single-variable calculus section? Thanks
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    Moving crests

    Homework Statement A crest is created on a string by vibrating it. The crests have two periods on the string which is 3 m long. At what point will the crest have an acceleration of zero? I had gotten this problem from a science competition and really had no idea how to do it. But I assumed...
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    Integral of sec(x)^3

    Homework Statement I'm just trying to solve this \int {\sec^3{x} dx Homework Equations sec^2{x} = 1 + tan^2{x} The Attempt at a Solution well i was able to simplify it to this: \int {\sec{x}*\tan^2{x}} dx + \ln{|\sec{x} + \tan{x}|} but I still was not able to find that new integral
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    Why is the electric field intensity zero at only one point?

    Suppose you have a Cartesian coordinate system with: a 2uC charge at (0,0) and a -8uC charge at (4,0), why is there only one point in which the EFI is zero (-4,0)? Shouldn't that be the first point where the EFI is zero (so (-5,0),(-6,0)... all have Electric Fields of zero) The reason I...