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Leibniz notation

  1. Dec 19, 2008 #1
    Hello I made a youtube video trying to explain Leibniz notation because it is something I found very difficult at first and very non intuitive. There doesn't seem to be many good explanations on the internet either, leading me to believe that no one fully understands the notation.

    I would like other people to take a look at it and make sure I'm not spreading any misinformation.

    My goal is not mathematical rigor, just to help people understand.

     
    Last edited by a moderator: Sep 25, 2014
  2. jcsd
  3. Dec 19, 2008 #2
    hey, your video seemed accurate to me. I think this is very helpful; for a long time in my highschool calculus class, I couldn't figure out what the hell dy/dx was (thinking of it as a function instead of a fraction...of sorts), then implicit differentiation came along and kicked my *** so hard it forced me to figure all this out on my own (my teacher wasn't much help and I don't understand math at all unless I understand it fully). So I, on behalf of all those who have struggled with this notation, thank you.
    By the way, I haven't thought of it much since back then but the slide at 0:21 is very helpful. I don't know if you have any experience with animation, but if you do, please elaborate to really get the images flowing. Good work.
    -Tim
     
  4. Dec 20, 2008 #3
    That is good to hear, I thought I was abusing the notation somewhat stating that dx/dx is 1 because I have NEVER seen anyone use this, despite the fact that mentioning it would make the notation a lot more consistent. Part of the reason I think implicit differentiation is so hard is because they make a distinction from the regular differentiation, when its basically the same thing only you don't have to re-arrange.
     
  5. Dec 20, 2008 #4
    Yea, that is EXACTLY what got me all mixed up with it. Once I thought of differentiating x with respect to x, it became much clearer. Even if this isn't exactly true, it should become so because it makes the notation very much easier to understand. Before noticing this, for a good two weeks I was very frustrated, going "ok, so this means put d_/dx next to the derivative everything that isn't in terms of x". After noticing dx/dx, I told a few of my classmates (they called me for help on a regular basis) and it also cleared things up for them as well. I got many "oooohhhhh"s.

    By the way, I think you should know that you have inspired me. Calculus does not have enough dynamic teaching tools. I'm taking the liberty of making them for Calc I. I'm working on part I (the derivative) right now, which analyzes the limit of f(x+h)-f(x)/h with only the assumption of the knowledge of algebra. I'm writing it in such a way that my mother would understand and not be frightened by the material (she's easily frightened by material that I know she would understand if she thought she could). For instance, I'll get to the limit later but for now I'm saying "let's just set h to some number, let's say 2". Later on, I'll show the points f(x+h) and f(x) getting dynamically closer to one another as the value listed next to "h" goes to zero. I realize this will not teach people how to properly evaluate a limit, but the concept is really all you need. I'm focusing on concepts rather than rigorous how-to's of algebraically manipulating formulas.

    Whatcha think?
     
  6. Dec 20, 2008 #5
    It is a good video.

    There is one book I could recommend to others:
    "CALCULUS An Intuitive and Physical Approach" by Morris Kline is good for those starting out (even if they have no beforehand knowledge of Calculus). It doesn't use modern notation too heavily, so its easy to follow through...its good stuff.
     
  7. Dec 20, 2008 #6

    jambaugh

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    I teach calculus at a liberal arts college and find that students often struggle with the concept of a differential. I hit them with these as early as I can so they have time to absorb it when we get to integration substitution at the end of the semester.

    Here's how I start, I use a chain of equivalences:

    [tex] m_{secant} = \frac{y_2 - y_1}{x_2 - x_1} = \frac{\Delta y}{\Delta x}[/tex]

    [tex]m_{tang.}=\lim m_{secant} =\lim_{\Delta x \to 0} \frac{\Delta y}{\Delta x} \equiv \frac{dy}{dx}[/tex]

    At this point I explain we use a fraction notation but this is not yet a fraction because we don't know what the numerator and denominator mean by themselves. I point out that by virtue of our limit laws this notation being a limit of a ratio will behave very much like a fraction. This helps when we then get to the chain rule.

    But I also mention that we will define something called differentials which are variables so that their ratio is the derivative represented in Leibniz's notation. In fact that is pretty much the definition right there. So this is indeed a fraction.

    It helps by the time I get to differentials. Unlike the text we use I get to differentials before I get to either related rates or implicit differentiation. This way we use (and practice) differentials while we work on these topics. It does daunt the students more at first but it also gives them more time, as I said, to absorb the concept before we get to integration.

    I'm still refining my approach but I find it a good one.
     
    Last edited: Dec 20, 2008
  8. Dec 20, 2008 #7
    We live in a great time where any random person has the tools to teach anyone else in the world how to do things in many different ways through sites like youtube. This gives great diversity to how the information that is out there is represented so that if you don't understand one explanation you can go on to the next. I think people often take for granted the fact that not everyone learns and relates information the same way they do, and tries to teach in a way that only certain people understand, but with the internet you can get as many different explanations as you want. Some people require just the algorithmic steps to understand things, but personally I need to know the physical interpretation before it sinks in.
     
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