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Math Help - leibniz notation

  1. #1
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    leibniz notation

    I have a question I should have asked a LONG time ago.

    When we see this notation being used in such formulas as i=dq/dt (definition of current) or dy/dx (etc, etc), are we saying (in the case of current) that current is equal to the change in charge with respect to time? Or is it the "current equal to the change in charge with respect to the change in time"?

    What if, hypothetically, the definition of current were written as i=d/dt. Does this mean "current is equal to the change in time"?

    I have always had issues reading/saying/comprehending leibniz notation. I know the calculus (meaning solving the problem), I just need to conceptualize what it represents. I hate solving a problem and NOT knowing what I'm doing.

    Any help is highly appreciated.
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    Intuitively, d is supposed to represent some arbitrarily small (or infinitesimal) quantity--so if we wrote i = d/dt it wouldn't have much of a meaning on its own. (It would obtain some significance if we multiplied i times some function which depends on time.) i = dq/dt expresses the first of the two concepts you mentioned, namely that i is equal to the change in current as time changes, for all real number values of time. Since change in current over some time intervals may be sharper than it is over some others, i is itself a function (not necessarily constant) of t.

    Now I'm not really sure how this concept is different from the second thing you wrote, "current equal to the change in charge with respect to the change in time". What is change in charge with respect to change in time except simply change in charge as time changes? Perhaps you have some sense of how this is different, which I'm not seeing.

    It is probably helpful to not think about Leibniz notation as anything other than a short-hand for the official definition of a derivative, which is the limit as h goes to zero of the difference quotient. (Note, a non-standard form of calculus has been developed which in some sense literally has infinitesimal quantities, and is provably equivalent to the calculus that you're learning. So there is a kind of legitimate way of thinking about infinitesimals, but since they're not formally introduced to you and you can do everything you have so far without them, you might just satisfy yourself with having an intuitive understand which--if you ever want to come back to a more formal treat--you can ultimately do away with and replace with the formal notion of a limit of the difference quotient.)

    Hope this helps.
    Last edited by ragnar; April 22nd 2011 at 01:31 PM. Reason: Mis-chosen word
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  3. #3
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    Quote Originally Posted by skaterbasist View Post
    I have a question I should have asked a LONG time ago.

    When we see this notation being used in such formulas as i=dq/dt (definition of current) or dy/dx (etc, etc), are we saying (in the case of current) that current is equal to the change in charge with respect to time? Or is it the "current equal to the change in charge with respect to the change in time"?
    "with respect to the change in time" but typically the "change in" is understood. (The rate of change of time is, after all, pretty constant.)

    What if, hypothetically, the definition of current were written as i=d/dt. Does this mean "current is equal to the change in time"?
    No, if you define current to mean non-sense, you get non-sense and it doesn't mean anything! "d/dt" is not a quantity and so cannot be set equal to a quantity like current. "d/dt" can be interpreted as an operator but it still cannot be equal to a quantity, such as current, until that operator is applied to something. If current were defined to be "d/dt" of some quantity, then it would be "rate of change" of that quantity with respect to the rate of change of time.

    I have always had issues reading/saying/comprehending leibniz notation. I know the calculus (meaning solving the problem), I just need to conceptualize what it represents. I hate solving a problem and NOT knowing what I'm doing.

    Any help is highly appreciated.
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    Thank you. It's making a little more sense. I have always found the 'prime' notation to be much easier to interpret, but I guess with physics and differential equations the leibniz notation is just more specific.

    For Math problems, if we were to see something as (dy/dx)(x+y)=(some constant), would that be the same as saying x'+y'=some constant? (Not sure if this made up problem makes any sense).
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    Quote Originally Posted by skaterbasist View Post
    Thank you. It's making a little more sense. I have always found the 'prime' notation to be much easier to interpret, but I guess with physics and differential equations the leibniz notation is just more specific.

    For Math problems, if we were to see something as (dy/dx)(x+y)=(some constant), would that be the same as saying x'+y'=some constant? (Not sure if this made up problem makes any sense).
    The main reason that Physics uses Leibniz notation is that there are too many variables to take derivatives of. (And Newton's notation is even worse for this.) For example, in Graduate level Mechanics it is not uncommon to see an equation that looks like something on the order of

    where the dot over the x represents a time derivative (Newton's notation) and the prime represents a derivative with respect to x. It gets very confusing very quickly. Leibniz notation is just simpler in such instances. And for partial differential equations Leibniz notation is practically a necessity.

    Your problem, by the way, is not correct. I suspect what you wanted to write was

    where a prime denotes a derivative with respect to t.

    -Dan
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    Quote Originally Posted by skaterbasist View Post
    Thank you. It's making a little more sense. I have always found the 'prime' notation to be much easier to interpret, but I guess with physics and differential equations the leibniz notation is just more specific.

    For Math problems, if we were to see something as (dy/dx)(x+y)=(some constant), would that be the same as saying x'+y'=some constant? (Not sure if this made up problem makes any sense).
    I wouldn't say Leibniz notation is more specific--they're equally mathematically rigorous. I would say that Leibniz notation has two advantages. First, it is easier to read, as Sir Toppy mentioned. (I'm sorry, but I might not be able to resist becoming increasingly silly with your name, and I know not why.) A second advantage is that it makes good physical sense. The difference quotient makes physical sense enough but when you see it in complicated equations, it's hard to get a gut feeling for what's going on. When you see deltas, you can just be like, "Oh, tiny change here over tiny change there is equal to this function." It makes it easy to think about what is changing with respect to what, and to use that information to understand the physical significance of equations.

    Actually, another good reason is because when you do differential equations you can treat dt like any other number and re-arrange it in an equation to solve. For instance, to solve the differential equation dy/dx = x you can multiply both sides of the equation by dx getting dy = xdx then integrate both sides, getting y = (x^2)/2 + C. Now in order to be assured that this solution is arrived at by valid reasoning, you must first see a proof that f(y)dy = g(x)dx implies int(f(y)dy) = int(g(x)dx), but such a proof exists. Writing all of this out in terms of the difference quotient would be a nightmare.
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