Results 1 to 10 of 10
Like Tree6Thanks
  • 2 Post By JJacquelin
  • 4 Post By HallsofIvy

Math Help - why can we treat dy/dx as variables?

  1. #1
    Junior Member
    Joined
    Dec 2012
    From
    ny
    Posts
    63
    Thanks
    6

    why can we treat dy/dx as variables?

    I've done some epsilon proofs in calculus, but in the Leibniz notation dy/dx why can we treat these as variables as when solving differential equations?
    Follow Math Help Forum on Facebook and Google+

  2. #2
    Member
    Joined
    Oct 2009
    Posts
    107
    Thanks
    2

    Re: why can we treat dy/dx as variables?

    I think its regarded as sloppy maths to do so and not technically correct
    Follow Math Help Forum on Facebook and Google+

  3. #3
    Junior Member EliteAndoy's Avatar
    Joined
    Feb 2013
    From
    United States
    Posts
    56
    Thanks
    3

    Re: why can we treat dy/dx as variables?

    Because they can be considered as a parameter since dy/dx is just the rate of change of y with respect to x.
    Follow Math Help Forum on Facebook and Google+

  4. #4
    Member
    Joined
    Aug 2011
    Posts
    244
    Thanks
    56

    Re: why can we treat dy/dx as variables?

    Hi !
    The question about dx and dy notation is an old well-worn subject.
    The "Nonstandard Analysis" theory leads to a strict answer. To French readers, "Une querelle des Anciens er des Modernes" : JJacquelin's Documents | Scribd
    Last edited by JJacquelin; September 1st 2013 at 11:30 PM.
    Thanks from emakarov and topsquark
    Follow Math Help Forum on Facebook and Google+

  5. #5
    Member
    Joined
    Sep 2012
    From
    Planet Earth
    Posts
    194
    Thanks
    49

    Re: why can we treat dy/dx as variables?

    dy and dx themselves are difficult to conceive of as variables. But \frac{dy}{dx} as a whole just means the derivative at a particular location, so it can be manipulated without requiring too much special care (until you try to separate the fraction).
    Follow Math Help Forum on Facebook and Google+

  6. #6
    MHF Contributor

    Joined
    Apr 2005
    Posts
    15,693
    Thanks
    1466

    Re: why can we treat dy/dx as variables?

    The derivative, dy/dx, is not defined as a fraction but it is defined as the limit of a fraction: \lim_{h\to 0} \frac{y(a+ h)- y(a)}{h}. One can show that "fraction properties" can be used by going back before the limit, applying the fraction property, then taking the limit. To make that a little more formal, Calculus texts then introduce the "differential", dx, separately from the derivative, then defining, dy, by "dy= (dy/dx) dx". ("Nonstandard Analysis" avoids that whole question but requires some very deep concepts in logic, introducing a completely different number system.)
    Follow Math Help Forum on Facebook and Google+

  7. #7
    Banned
    Joined
    Aug 2010
    Posts
    961
    Thanks
    98

    Re: why can we treat dy/dx as variables?

    f(x) has a standard definition not involving dy and dx.

    dy=f(x)dx is a defined relation between the variables dy and dx.

    For example, if f(x)=5, dy = 5dx by definition. If dx is 10, dy=50. If dy is 10, dx=2.

    dy and dx can be interpreted as increments on the tangent line.
    Last edited by Hartlw; September 3rd 2013 at 09:38 AM.
    Follow Math Help Forum on Facebook and Google+

  8. #8
    Junior Member
    Joined
    Dec 2012
    From
    ny
    Posts
    63
    Thanks
    6

    Re: why can we treat dy/dx as variables?

    \frac{dy}{dt}F(y(t))=g(t),\int_{to}^{t}\frac{dy}{d  t}F(y(t))=\int_{to}^{t}g(t)=F(y(t))-F(yo), \int_{t}^{to}f(r)dr=\int_{to}^{t}g(s)ds,
    in differential equations when solving an initial value problem for c, you have this proof in Braun where he changes f(t) to f(r), g(t) to g(s)- I'm not sure what the reasoning is, maybe you can use any arbitrary variable since when you integrate the limits will give you dependence on t an to? the time dependence of y on t also remains? and then it's assumed you can integrate both sides
    Last edited by mathlover10; September 5th 2013 at 10:09 PM.
    Follow Math Help Forum on Facebook and Google+

  9. #9
    Junior Member
    Joined
    Dec 2012
    From
    ny
    Posts
    63
    Thanks
    6

    Re: why can we treat dy/dx as variables?

    what are some texts on non standard analysis and logic. I really love Tarski and modal logic and wish I could be a mathematician
    Follow Math Help Forum on Facebook and Google+

  10. #10
    Junior Member
    Joined
    Dec 2012
    From
    ny
    Posts
    63
    Thanks
    6

    Re: why can we treat dy/dx as variables?

    so when you integrate both sides is the process going to be exactly like balancing arithmetic? you can really choose any arbitrary function taking the inverse mapping?
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. How to treat dy/dx when integrating
    Posted in the Differential Equations Forum
    Replies: 1
    Last Post: August 28th 2013, 03:03 PM
  2. Replies: 4
    Last Post: June 27th 2012, 05:46 AM
  3. Replies: 1
    Last Post: October 11th 2011, 02:29 AM
  4. Replies: 1
    Last Post: June 5th 2011, 03:57 PM
  5. solve variables in terms of variables
    Posted in the Algebra Forum
    Replies: 4
    Last Post: February 5th 2007, 10:46 PM

Search Tags


/mathhelpforum @mathhelpforum