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B How is this mathematically rigorous?

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  1. Mar 8, 2017 #1
    IMG_20170308_231451_01.jpg
    In the solution of this problem(121), dW = (kmgcosΦ + mgsinΦ) ds, where ds is the differential element along the curve. Now they have done kmg dscosΦ + mg ds(sinΦ) = kmgdx +mgdy. Makes sense intuitively, but I want to know how this is rigorous. What I'm thinking is, the curve is broken into N elements of the same length along the curve over which a Riemann sum gives a line integral. But here cosΦ is varying, so wouldn't all the Δx's and Δy's not be of the same length? And if your explanation is that as Δx tends to zero the two become the same, then why can't we simply treat all differential elements as the same since they all tend to zero after all?

    Sorry for the rambling at the end. But can someone just show me why such an operation is rigorous?
     
  2. jcsd
  3. Mar 8, 2017 #2

    jedishrfu

    Staff: Mentor

    Isn't ##ds*cos(\phi) = dx## and ##ds*sin(\phi) = dy## ?

    You're breaking the ##ds## into components along x and y directions.
     
  4. Mar 8, 2017 #3
    the curve is broken into N elements of the same length along the curve over which a Riemann sum gives a line integral. But here cosΦ is varying, so wouldn't all the Δx's and Δy's not be of the same length?
     
  5. Mar 8, 2017 #4

    Mark44

    Staff: Mentor

    No, they wouldn't. In portions of the curve where the angle is less than 45°, ##\Delta x## will be larger than ##\Delta y##. In other portions, where the angle is greater than 45°, ##\Delta y## will be larger. The curve is broken up into equal-length segments, with each segment being approximately the hypotenuse of a right triangle. Even though these segments of arc length are the same size, along the curve, that doesn't imply that the horizontal and vertical legs of the triangles will be equal in length.
     
  6. Mar 8, 2017 #5
    I think you misread what I posted. I posted: "But here cosΦ is varying, so wouldn't all the Δx's and Δy's not be of the same length?"

    Now since the Δx's and Δy's everywhere would vary with Φ, how can we do a Riemann sum along x and y when the interval that we're using itself is varying?
     
  7. Mar 9, 2017 #6

    Mark44

    Staff: Mentor

    In a Riemann sum the intervals along the curve don't have to be the same size, but as the number of subintervals increases, the lengths of ##\Delta s##, ##\Delta x##, and ##\Delta y## grow smaller as well, but again, that doesn't mean they all are the same size.
     
  8. Mar 9, 2017 #7
    But here if Δx is changing Φ with and we don't know the curve either, we can't do Δx = l/N. How would we evaluate this Riemann sum?
     
  9. Mar 9, 2017 #8

    Mark44

    Staff: Mentor

    If you know ##\Delta s## and ##\phi##, you can get ##\Delta x## and ##\Delta y## -- simple trig.
     
  10. Mar 9, 2017 #9
    But we don't know how Φ varies since we don't know the curve
     
  11. Mar 9, 2017 #10

    Mark44

    Staff: Mentor

    In the picture, there is a force ##\vec{F}## that is always tangent to the curve. That's enough to get you the angle ##\phi##, since ##\phi## is the angle between ##\vec{F}## and the horizontal. Although not stated, I believe the assumption is that the direction ##\vec{F}## points would be known at each point along the curve.
     
  12. Mar 9, 2017 #11
    How would we know that if we don't know the curve?
     
  13. Mar 9, 2017 #12

    Mark44

    Staff: Mentor

    Assume that the curve is given by y = f(x), which you can assume to be a differentiable function. At any point (x, f(x)) on the curve, you can find the tangent to the curve, right? I.e., at the point (x, f(x)), the slope of the tangent line is f'(x). From that you can write an expression for the angle ##\phi##.
     
  14. Mar 9, 2017 #13

    Mark44

    Staff: Mentor

    Before going any further, what's your background in mathematics? IOW, what classes have you taken?
     
  15. Mar 9, 2017 #14
    I know single variable calculus from math but I do know a bit about line integrals, and a little bit of vector calculus because of the physics I'm trying to learn.
     
  16. Mar 9, 2017 #15

    Mark44

    Staff: Mentor

    OK, so do you understand what I write in post #12?
     
  17. Mar 9, 2017 #16

    Demystifier

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    Let me try to anticipate what he will say next: "Yes, but how can I know that f(x) is differentiable if I don't know what f(x) is?"
     
  18. Mar 9, 2017 #17
    We can write f'(x)= tanΦ but we don't know f(x).
     
  19. Mar 9, 2017 #18

    Demystifier

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    I was close. :biggrin:

    You need to think in an abstract way. That allows you to say a lot about things even if you don't know what they are. For instance one grbljh + one grbljh equals two grbljh even if you have no idea what grbljh is.
     
  20. Mar 9, 2017 #19
    Haha :p
     
  21. Mar 9, 2017 #20

    Demystifier

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    Seriously, the point is that if something is true for any f(x), then you don't need to know what is f(x) to make a rigorous argument about f(x).
     
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