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Thread: Several Variable Calculus

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    Ancient Orator leehuan's Avatar
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    Several Variable Calculus

    For some reason I lost the point of intersection (2,0):



    I equated the relevant r1 and r2 components but only got t=0

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    Re: Several Variable Calculus

    Quote Originally Posted by leehuan View Post
    For some reason I lost the point of intersection (2,0):



    I equated the relevant r1 and r2 components but only got t=0
    Solve the system of equations

    t^2 - t = s + s^2 (1)
    t^2 + t = s - s^2 (2).
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    Re: Several Variable Calculus

    Quote Originally Posted by InteGrand View Post
    Solve the system of equations

    t^2 - t = s + s^2 (1)
    t^2 + t = s - s^2 (2).
    Oh. I'll work on that right now but why was it necessary to introduce a new variable?

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    Re: Several Variable Calculus

    Quote Originally Posted by leehuan View Post
    Oh. I'll work on that right now but why was it necessary to introduce a new variable?
    You have two parametric curves. Their points of intersection don't necessarily have the same parameter as a point on the first curve as they do as a point on the second curve.
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    Re: Several Variable Calculus

    Eg, consider the lines (x,y)=(t,0), (x,y)=(2t,0).

    These lines coincide exactly, so every point on the x-axis is a point of intersection.

    Yet the only place where (t,0)=(2t,0) is at the origin.
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    Re: Several Variable Calculus

    Excellent. Makes sense.
    ________________

    Find the angle between the two curves at the points of intersection.

    I'm having a dumb moment now. Which vectors are we taking the dot product of?

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    Re: Several Variable Calculus

    Quote Originally Posted by leehuan View Post
    Excellent. Makes sense.
    ________________

    Find the angle between the two curves at the points of intersection.

    I'm having a dumb moment now. Which vectors are we taking the dot product of?
    Find the s and t values at the points of intersection and plug them into the derivatives of the parametric curves. This will give us the "direction vectors" of the curves at the points of intersection. Find the angle between these vectors.
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    Re: Several Variable Calculus

    Strange... That's what I did so maybe there's an error in my computation.









    But the answer was arccos(0.8)

    EDIT: Ouch. I know what I did now.
    Last edited by leehuan; 28 Feb 2017 at 12:31 PM.

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    Re: Several Variable Calculus

    Last edited by InteGrand; 28 Feb 2017 at 12:32 PM. Reason: Typo
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    Re: Several Variable Calculus

    Quote Originally Posted by leehuan View Post
    Strange... That's what I did so maybe there's an error in my computation.









    But the answer was arccos(0.8)
    You appear to have miscalculated the velocity vectors when subbing in the values of t and/or s (check the first components, noting you're subbing in s = 1 (not 2) and t = -1 (not -2)).
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    Re: Several Variable Calculus







    Can be assumed: F=ma

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    Re: Several Variable Calculus

    Quote Originally Posted by leehuan View Post






    Can be assumed: F=ma
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    Re: Several Variable Calculus

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    Re: Several Variable Calculus

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

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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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    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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    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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    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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    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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    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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    Re: Several Variable Calculus




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