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Thread: MATH2111 Higher Several Variable Calculus

  1. #76
    Ancient Orator leehuan's Avatar
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    Re: Several Variable Calculus

    Quote Originally Posted by InteGrand View Post
    Completely forgot about this haha

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    Ancient Orator leehuan's Avatar
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    Re: Several Variable Calculus






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    Re: Several Variable Calculus

    Quote Originally Posted by leehuan View Post




    Here's some hints.



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    Ancient Orator leehuan's Avatar
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    Re: Several Variable Calculus

    Quote Originally Posted by InteGrand View Post
    Here's some hints.



    Is it basically just this?



    Although that being said I had c_1 instead of c_2 but I feel that won't matter
    Last edited by leehuan; 4 Mar 2017 at 8:35 PM.

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    Re: Several Variable Calculus




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    Re: Several Variable Calculus

    Quote Originally Posted by leehuan View Post




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    Ancient Orator leehuan's Avatar
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    Re: Several Variable Calculus

    Quote Originally Posted by InteGrand View Post


    Appears so counterintuitive though. I can't visualise what's going on here

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    -insert title here- Paradoxica's Avatar
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    Re: Several Variable Calculus

    Quote Originally Posted by leehuan View Post
    Appears so counterintuitive though. I can't visualise what's going on here
    nobody can.....
    If I am a conic section, then my e = ∞

    Just so we don't have this discussion in the future, my definition of the natural numbers includes 0.

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    Cadet QuantumRoulette's Avatar
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    Re: Several Variable Calculus

    Consider the two metrics and
    (you may assume they are metrics).
    i) Show that d and δ are not equivalent.
    Last edited by QuantumRoulette; 7 Mar 2017 at 10:30 PM. Reason: fixed Tex issues

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    Supreme Member seanieg89's Avatar
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    Re: Several Variable Calculus

    Quote Originally Posted by QuantumRoulette View Post
    Consider the two metrics and
    (you may assume they are metrics).
    i) Show that d and δ are not equivalent.
    I assume d(x,y) is supposed to be ||x-y|| and the set is some normed vector space V with norm ||.|| (e.g. R^d). (Please specify more if this is not the intended setting.)

    Then these two metrics are not (strongly) equivalent because V is bounded with the delta metric but unbounded with the d metric.

    That V is bounded with the delta metric follows immediately from the definition of delta, which must always lie in [0,1). On the other hand d(tx,0)=|t|d(x,0) can be made arbitrarily large for nonzero x.


    Note that these two metrics ARE topologically equivalent though, in the sense that convergence in one metric implies convergence in the other. This follows from from the map x->x/(1+x) being a homeomorphism from [0,inf) to [0,1).
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    Supreme Member seanieg89's Avatar
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    Re: Several Variable Calculus

    Quote Originally Posted by leehuan View Post
    Appears so counterintuitive though. I can't visualise what's going on here
    Might not be easy to visualise the graph of the function on all of R^2, but you should certainly be able to visualise what it looks like on the slices x=const. or y=const which is all that matters for seeing/proving the nonexistence of the iterated limit. It is an oscillatory expression that oscillates faster as you approach axes. One of the two summands becomes irrelevant as you get close to the axes, so the other one dominates. This thing behaves like (const).sin(1/x), which of course does not converge unless that const is zero.

    The boundedness of sine makes it clear that f(x,y) tends to zero as (x,y) tends to zero though.

    Long story short: don't be too hasty to form intuitions in analysis, lots of things can behave weirdly...you kind of have to slowly build up a list of things that ARE true (via proof!) rather than assuming innocuous statements are true and ruling these things out as you come across pathological counterexamples.

    And when you are looking at functions like this, try to isolate the terms that actually matter for the property you are trying to prove. A large part of analysis is just approximating ugly things by nice things, throwing away small sets on which a function behaves badly, etc etc.
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    Ancient Orator leehuan's Avatar
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    Re: Several Variable Calculus




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    Re: Several Variable Calculus

    Quote Originally Posted by leehuan View Post






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    Supreme Member seanieg89's Avatar
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    Re: Several Variable Calculus

    Quote Originally Posted by leehuan View Post


    |sin(x)/x-1|=|x|^2 ?

    Surely you mean something like:

    |sin(x)/x-1| = O(|x|^2).

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    Ancient Orator leehuan's Avatar
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    Re: Several Variable Calculus

    Quote Originally Posted by seanieg89 View Post
    |sin(x)/x-1|=|x|^2 ?

    Surely you mean something like:

    |sin(x)/x-1| = O(|x|^2).
    There was a typo; it was meant to be a less than or equals to. I think InteGrand just ignored or figured I had typo'd and just used the correct inequality.
    __________________________________________________ ______________





    As in, for every ball around 0 there's always an irrational number in it, therefore 0 is not an interior point so it cannot be open. (I hope I did not screw this up.)
    ____________________________





    I wanted to use the floor/ceiling functions but then I realised that'd be problematic if r = 0.1, so what might be a good choice for a rational number that is in every ball B(sqrt2, r)?


    (Sorry, I think I worded my question horribly)
    Last edited by leehuan; 11 Mar 2017 at 11:26 AM.

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    Re: Several Variable Calculus

    Quote Originally Posted by leehuan View Post
    There was a typo; it was meant to be a less than or equals to. I think InteGrand just ignored or figured I had typo'd and just used the correct inequality.
    __________________________________________________ ______________





    As in, for every ball around 0 there's always an irrational number in it, therefore 0 is not an interior point so it cannot be open. (I hope I did not screw this up.)
    ____________________________





    I wanted to use the floor/ceiling functions but then I realised that'd be problematic if r = 0.1, so what might be a good choice for a rational number that is in every ball B(sqrt2, r)?


    (Sorry, I think I worded my question horribly)
    Some rational points arbitrarily close to sqrt(2) are points where we truncate the decimal expansion of sqrt(2) arbitrarily far. I.e.



    for positive integers n.
    Last edited by InteGrand; 11 Mar 2017 at 11:59 AM.
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    Ancient Orator leehuan's Avatar
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    Re: Several Variable Calculus

    Quote Originally Posted by InteGrand View Post
    Some rational points arbitrarily close to sqrt(2) are points where we truncate the decimal expansion of sqrt(2) arbitrarily far. I.e.



    for positive integers n.
    That is one beautiful trick.
    _____________________________________



    I definitely don't want to use the Lagrange multiplier theorem to find the closest point (x,y) to work off, so any pointers on what radius I should choose my ball to be? (to prove an arbitrary point (x,y) is interior)

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    Re: Several Variable Calculus

    Quote Originally Posted by leehuan View Post
    That is one beautiful trick.
    _____________________________________



    I definitely don't want to use the Lagrange multiplier theorem to find the closest point (x,y) to work off, so any pointers on what radius I should choose my ball to be? (to prove an arbitrary point (x,y) is interior)
    That set Omega is an open ball already (using the Euclidean metric), and an open ball is an open set (you should prove this as an exercise if you haven't before).
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    Re: Several Variable Calculus

    Mate, stop cheating on these assignment questions. Do it yourself ffs.

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    Ancient Orator leehuan's Avatar
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    Re: Several Variable Calculus

    Quote Originally Posted by MATH2111 View Post
    Mate, stop cheating on these assignment questions. Do it yourself ffs.
    ...?

    These are homework questions I'm stuck on. I'd be too worried about getting a 0 mark for the course if these were assignment questions.

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    Ancient Orator leehuan's Avatar
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    Re: Several Variable Calculus

    Quote Originally Posted by InteGrand View Post
    That set Omega is an open ball already (using the Euclidean metric), and an open ball is an open set (you should prove this as an exercise if you haven't before).
    Wait, I can't see it - Can't tell how a region cut off by the rectangular hyperbola is a ball

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    Re: Several Variable Calculus

    Quote Originally Posted by leehuan View Post
    Wait, I can't see it - Can't tell how a region cut off by the rectangular hyperbola is a ball
    Sorry misread it (saw + instead of -).

    One way is to recall that the preimage of an open set under a continuous map is open.
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    Ancient Orator leehuan's Avatar
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    Re: Several Variable Calculus




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    Re: Several Variable Calculus

    Quote Originally Posted by leehuan View Post










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    Re: Several Variable Calculus

    Just a brief sketch-out please





    I know that f is not continuous at 0 but I'm not sure if that helps since we're talking about partial derivatives here.

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