Fundamental confusions of calculus

Discussion in 'Physics & Math' started by arfa brane, Feb 11, 2012.

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  1. Trippy ALEA IACTA EST Staff Member

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    I suppose that depends on your thought processes really, where it might confuse some people, to me it emphasizes the point :shrugs:.

    I can imagine that some people might, sure. But if people prefer to use a and a[sup]2[/sup] that really doesn't bother me in the slightest because as long as one remembers that b=a[sup]2[/sup]=y[sup]2[/sup] it's fine, we're still talking about the same thing.

    I know, it's not something that I would ordinarily use either, I was just using it to emphasize a point (or try to anyway).
     
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  3. Tach Banned Banned

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    The only issue is that you failed to connect the theoretical part in post 18 with the example. This boils down to your failing to identify \(u\). This is what I call arguing in bad faith.
     
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  5. Trippy ALEA IACTA EST Staff Member

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    You mean the example I bought up?

    That example?

    You're asking me if I remember a post I made in this thread?

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    I'm well aware of that point, but apparently you're ignoring mine, or it went over your head.

    You have on three seperate occasions:
    Left u undefined explicitly.
    Defined u as being sin(\(\theta\))
    Defined u as being sin[sup]2[/sup](\(\theta\))
     
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  7. przyk squishy Valued Senior Member

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    Once again, the material I cited had nothing to do with the definition of \(u\).

    You are deflecting criticism of real and unambiguous errors you made. That is bad faith.
     
  8. Tach Banned Banned

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    Do you understand that there is no difference between the two cases? They produce the same partial and total derivative. Can you prove it?
     
  9. Tach Banned Banned

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    You mean the imagined errors resulting from your failure to identify \(u\) and to apply the derivative only to \(3 \theta\) (because \(sin^2(\theta)=u)\). Even after being told that to be the case for about 11 times?
    Let me ask you something:

    \(z=3 \theta +u+v\)
    \(u=sin^2(\theta)\)
    \(v=x^6\)

    What is \(\frac{\partial z}{\partial \theta}\)?
    What is \(\frac{d z}{d \theta}\)?
     
    Last edited: Feb 15, 2012
  10. Trippy ALEA IACTA EST Staff Member

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    I don't have anything to prove in this thread. You do.
     
  11. Tach Banned Banned

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    I would hope that you understand that there is no difference between the two examples. So, why do you bring up the issue?
     
  12. przyk squishy Valued Senior Member

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    No. [POST=2904021]Learn to read[/POST]. I mean these very real errors, which are independent of how \(f\) and \(u\) are defined:

    The first two words say a lot when put in the context of your entire post:
    You believed the part I quoted above was true in general, and independent of how \(f\) and \(u\) were defined. So all this stuff about how \(u\) and \(f\) are defined is pathetic deflection on your part.
     
  13. Tach Banned Banned

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    So, you wouldn't have any difficulty in answering:


    \(z=3 \theta +u+v\)
    \(u=sin^2(\theta)\)
    \(v=x^6\)

    What is \(\frac{\partial z}{\partial \theta}\)?
    What is \(\frac{d z}{d \theta}\)?

    Very simple questions.
     
  14. przyk squishy Valued Senior Member

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    No, I wouldn't have any difficulty answering any of these questions, and nor would anyone else participating in this thread, and you know it. What does that have to do with anything?

    Didn't I just say multiple times that the errors I was pointing out had nothing to do with how \(f\) and \(u\) were defined?

    If you'd admit you made these errors, that'd be great. If you'd rather silently acknowledge them by ceasing to reply to my posts on that subject, then I'd also consider that acceptable, as long as you don't repeat the reason I brought them up in the first place (explained [POST=2904009]here[/POST]). What's really not acceptable is deflection.
     
  15. Pete It's not rocket surgery Moderator

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    Actually, I think that people would have difficulty with \(dz/d\theta\), considering that he's asking for the total derivative of a multivariable function (assuming x is a variable, not an oddly named constant).
    Trick question?
    Or is Tach still confused?
     
  16. Trippy ALEA IACTA EST Staff Member

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    Because of your insistence that you defined u in the first place, which you didn't, followed by your insistence that you defined u as being a particular thing, when you've defined it as being two different things.
     
  17. Pete It's not rocket surgery Moderator

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    Notation related question for the educated.

    Regarding the above, if I wrote:
    \(\frac{\partial}{\partial \theta}z(\theta, u(\theta), v(x))\)​
    ...would that be interpreted as equivalent to
    \(\frac{\partial}{\partial \theta}z(\theta,u,v) = \frac{\partial}{\partial \theta}(3\theta + u + v)\)​
    ...or...
    \(\frac{\partial}{\partial \theta}Z(\theta, x) = \frac{\partial}{\partial \theta}(3\theta + sin^2\theta + x^6)\)​
    ...or is it just poor notation?
     
  18. arfa brane call me arf Valued Senior Member

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    To me, that says: "take the partial derivative of z with respect to θ, with x constant"
    That looks like a different function that doesn't define arguments u or v as functions but variables.
    That looks like it has the same form as the first function but with u(θ) and v(x) explicitly defined. What you haven't done is write the first function as an equation.

    In each case you get a different answer, though.
     
  19. Pete It's not rocket surgery Moderator

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    Thanks, I think I've got a good handle on it now.
    That's why I'm asking, and where I think Tach is confused.

    Given:
    \(z = Z(\theta, \, u, \, v) \\ u = U(\theta) \\ v = V(x) \\ Z_1(\theta,x) = Z(\theta, \, U(\theta), \, V(x))\)​

    Then I think that Tach is misnaming \(\partial Z_1/\partial x\) as a total derivative \(dz/d\theta\).
    In post 210, the answer he is expecting for \(dz/d\theta\) is \(3 + \sin(2\theta)\), which is of course the partial derivative \(\frac{\partial Z_1}{\partial \theta}\) for \(Z_1(\theta, x) = 3\theta + \sin^2\theta + x^6\)

    If that is what he's thinking, then I can certainly understand the mistake. I've learned a lot in this thread (and still learning) about the dangers of playing fast and loose with notation when playing with functions and their parameters.
    (I hope that the notation in this post is sufficiently close to good practice!)
     
    Last edited: Feb 15, 2012
  20. temur man of no words Registered Senior Member

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    My answer is the last two..
     
  21. temur man of no words Registered Senior Member

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    Yes. This is what he is thinking. It doesn't help that a partial derivative is just the "total" derivative along one of the coordinate axes. In fact I would recommend forgetting altogether about "total" derivative. There are only partial derivatives for multivariate functions (more precisely, only the directional derivatives). "Total" derivative is just the ordinary derivative of a single variable function that is constructed out of a multivariate function in a certain way.
     
    Last edited: Feb 15, 2012
  22. Pete It's not rocket surgery Moderator

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    Thought so, thanks

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    ...so rather than talking about the total derivative of a multivariable function, it's better to explicitly define the related single variable function, and talk about that derivative.

    For example, in the infamous bug-on-a-hotplate video, Metzler describes functions:
    \(T = f(x,y,t)\) (the temperature of the hotplate at given coordinates at a given time)​
    \(x = g(t) \\ y = h(t)\) (the path coordinates of a bug walking on the hotplate)​

    He then describes using the total derivative dT/dt to determine the rate of change in temperature felt by the bug on its walk across the plate.

    But!
    It would be better (would have saved much confusion!) if he had instead described a separate function for the temperature felt by the bug:
    \(\begin{align} T_{\tiny{bug}} &= F(t) \\ &= f(g(t), h(t), t) \end{align}\)​
    and talked about \(\frac{dT_{\tiny {bug}}}{dt}\) instead.
    But then again, the purpose of the video was to explain the concept of a total derivative, so perhaps not.

    Questions...
    • in the previous post I labelled two functions \(Z\) and \(Z_1\).
      Is it Ok to use a numeric subscript to simply distinguish two different but related functions, or does a subscript imply something else?
    • Is it OK to use a F(t) for a path-function (or whatever you call it) through f(x,y,t), or (thinking back to high school) does F(t) imply the antiderivative of f(t)?
    • If you have an expression \(y = f(x)\), is it preferable to use \(\frac{dy}{dx}\) or \(\frac{df}{dx}\) or \(\frac{d}{dx} f(x)\) to denote the derivative?
      I'm guessing it's best to leave y out altogether, and just define f(x) in the first place.
     
    Last edited: Feb 15, 2012
  23. temur man of no words Registered Senior Member

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    • It is ok. Unless.. If Z is a vector, subscript may indicate its particular component.
    • Ok. If you need F to be an antiderivative you explicitly say it.
    • All fine. Strictly speaking, the notation \(\frac{d}{dx} f(x)\) means that you take the derivative, and then evaluate it at point x.
     
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