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Thread: Splitting is cheating?

  1. #1 Splitting is cheating? 
    Forum Freshman Futurist's Avatar
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    Why do all of my maths lecturers tell me that separating derivatives is bad?
    like:

    dx/dt =1
    dx = dt

    this is usually followed by integration. We do it all the time in physics, but they tell me that physics is a sloppy discipline :?

    oh, and newbie question: what equation editor do you forumers use here?


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  3. #2 Re: Splitting is cheating? 
    . DrRocket's Avatar
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    Quote Originally Posted by Futurist
    Why do all of my maths lecturers tell me that separating derivatives is bad?
    like:

    dx/dt =1
    dx = dt

    this is usually followed by integration. We do it all the time in physics, but they tell me that physics is a sloppy discipline :?

    oh, and newbie question: what equation editor do you forumers use here?
    They are right.

    The reason that they tell you that dx/dt=1 should not be converted to dx=dt is because in calculus classes the expression dx has no meaning. dx/dt is not a fraction.

    There are ways to do this rigorously, but in calculus classes you are not prepared to discuss them. And even if at some point you get to the point where it starts to make sense, derivatives are still not a fraction.

    Physicists do mathematical manipulations that often do not make sense. Sometimes they get away with it, largely because there are means that they don't understand whereby that manipulation can be justified in a larger context. But sometimes they just get nonsense. And often they don't know which without a physical experiment.


    In the case of dx/dt =1. You know that then x =t + constant. So you formally use
    dx or dt interchangeable in an integral, as it is just the change of variables process that you learned in calculus.

    So you get to something that is correct by a process that is on the surface quite nonsensical. Basically compensating errors.

    But also as I said there is a way to do this with more rigor, but to do that you have to have a real definition for what dx means by itself. For that you will have to wait until you have quite a bit more mathematics under your belt.

    You might at some pont want to pick up Michael Spivak's litter paperback book Calculus on Manifolds where you can see the use of "dx" much more rigorously. You can read that book once you have pretty well mastered calculus and are ready for a modern treatment of calculus of several variables, or what is commonly called "advanced calculus". A good time to do that is right after you have taken a first course in electromagnetics and have seen some vector analysis (div,grad, curl etc and what are called the Divergence Theorem and Stokes Theorem in those classes). Spivak will show you that the Divergence Theorem, Stokes Theorem and the Fundamental Theorem of Calculus are really the same thing and can be generalized to what mathematicians all call just Stokes Theorem and that it applies in all dimensions.


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  4. #3 Re: Splitting is cheating? 
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    Quote Originally Posted by Futurist
    Why do all of my maths lecturers tell me that separating derivatives is bad?
    like:

    dx/dt =1
    dx = dt

    this is usually followed by integration. We do it all the time in physics, but they tell me that physics is a sloppy discipline :?

    oh, and newbie question: what equation editor do you forumers use here?
    The reason is that, algebraically, one cannot conduct operations on the notation of a derivative. A better way to view it is, for dy/dx, consider dy as the change in y of the tangent line to a point at x_0, and dx as the change in x of the tangent line to a point x_0. The notation consisting of deltas should not be viewed as the tangent line, but the function itself. It makes sense than that for values where the change in x is small that the derivative will be approximately equal to the tangent line.

    The only reason that the notation for a derivative takes the form of a fraction is because it represents a slope, and therefore Leibniz knew that providing it with a notation of differentials would suffice to express its meaning. Similarily, the integral of a function represents an area, or a product of two numbers, so it would make sense than that Leibniz would represent the notation as a product of the function and a differential (generally dx).
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  5. #4 Re: Splitting is cheating? 
    Forum Freshman Futurist's Avatar
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    Quote Originally Posted by DrRocket
    They are right.

    The reason that they tell you that dx/dt=1 should not be converted to dx=dt is because in calculus classes the expression dx has no meaning. dx/dt is not a fraction.

    There are ways to do this rigorously, but in calculus classes you are not prepared to discuss them. And even if at some point you get to the point where it starts to make sense, derivatives are still not a fraction.

    Physicists do mathematical manipulations that often do not make sense. Sometimes they get away with it, largely because there are means that they don't understand whereby that manipulation can be justified in a larger context. But sometimes they just get nonsense. And often they don't know which without a physical experiment.


    In the case of dx/dt =1. You know that then x =t + constant. So you formally use
    dx or dt interchangeable in an integral, as it is just the change of variables process that you learned in calculus.

    So you get to something that is correct by a process that is on the surface quite nonsensical. Basically compensating errors.

    But also as I said there is a way to do this with more rigor, but to do that you have to have a real definition for what dx means by itself. For that you will have to wait until you have quite a bit more mathematics under your belt.

    You might at some pont want to pick up Michael Spivak's litter paperback book Calculus on Manifolds where you can see the use of "dx" much more rigorously. You can read that book once you have pretty well mastered calculus and are ready for a modern treatment of calculus of several variables, or what is commonly called "advanced calculus". A good time to do that is right after you have taken a first course in electromagnetics and have seen some vector analysis (div,grad, curl etc and what are called the Divergence Theorem and Stokes Theorem in those classes). Spivak will show you that the Divergence Theorem, Stokes Theorem and the Fundamental Theorem of Calculus are really the same thing and can be generalized to what mathematicians all call just Stokes Theorem and that it applies in all dimensions.
    Thanks Dr. Rocket,

    I just started Advanced Calc. which we call Multivariable and complex Calculus, electromagnetism is coming up soon too. At the end of this course, there will be no more pure maths. At what point can you really say that you've mastered calc.???

    I looked this up on the internet a while back and a site was telling me that dx was an near infinitely small segment of x. How far is this off of the definition that you encounter towards the end of calculus?
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  6. #5 Re: Splitting is cheating? 
    . DrRocket's Avatar
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    [quote="Futurist"]
    Quote Originally Posted by DrRocket

    Thanks Dr. Rocket,

    I just started Advanced Calc. which we call Multivariable and complex Calculus, electromagnetism is coming up soon too. At the end of this course, there will be no more pure maths. At what point can you really say that you've mastered calc.???

    I looked this up on the internet a while back and a site was telling me that dx was an near infinitely small segment of x. How far is this off of the definition that you encounter towards the end of calculus?
    The definition of dx as an "infitesimal" goes back to Leibniz and is troublesome in rigorous mathematics. It is, however, common in elementary calculus (if a definition is provided at all).

    To see differentials done in a more rigorous and useful manner, you would need to take a class in which the notion of differential forms is discussed. This is sometimes done in classes with a title like "advanced calculus" but not in all such classes. You can find a good introductory treatment in Mike Spivak's little paperback Calculus on Manifolds which is at the level of advanced calculus and very readable.

    The theory of differential forms is part of the general subject of differential geometry, which is becoming more and more important to physics. You may therefore also be introduced to this subject in an advanced physics class, but I doubt at an undergraduate level. You would see it in many texts on general relativity (for instance Misner, Thorne and Wheeler's Gravitation) and some texts on mathematical physics (e.g. Thirring's A Course in Mathematical Physics).

    As far as at what level you can say that you have "mastered calc" I am not sure how to answer. Calculus is basically the first class that you take in the branch of mathematics called "analysis". Analysis is a very large subject. You can study it for a lifetime. I think there are probably Fields Medalists who would not claim to have mastered all of it. On the other hand I dislike the word "mastery" and generally feel that anyone who thinks that he has completely mastered any subject that has depth simply does not understand the problem or what he doesn't know.
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