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Integral total and partial of a function?

  1. Mar 31, 2014 #1
    Like we have the total differential of a function:
    imagem.png

    I was thinking, why not take the "total integral" of a function too? Thus I did some algebraic juggling and, how I haven't aptitude for be a Ph.D. in math, I bring my ideia for the experients from here evaluate... Anyway, the ideia is the follows:

    Let y = f(x), so: [tex]\int y dx = \int f dx[/tex] [tex]\int y \frac{dx}{dx} = \int f \frac{1}{dx}dx[/tex] [tex]\int y = \int f \frac{1}{dx}dx \;\;\;\Rightarrow \;\;\; \int y du = \int f \frac{du}{dx}dx[/tex] Generalizing...

    Let w = f(x,y,z), so: [tex]\int w = \int f \frac{1}{dx}dx + \int f \frac{1}{dy}dy + \int f \frac{1}{dz}dz[/tex]

    I don't venture take the 2nd integral of y because I think that will arise one d²x in the denominator...

    What you think about? All this make sense?
     
  2. jcsd
  3. Mar 31, 2014 #2

    Mark44

    Staff: Mentor

    Presumably dx2 means dx * dx, but what does d2x mean?
    The steps in the middle make no sense to me. Dividing by dx is not a valid step. You started with f as a function of x. Is it somehow transformed to become a function of u later on?
     
  4. Mar 31, 2014 #3
    Yeah, dx²=dxdx

    d²x is the 2nd differential of x wrt nothing. Wrt to something it's become: [tex]\frac{d^2y}{du^2}=\frac{d^2f}{dx^2}\left ( \frac{dx}{du} \right )^2+\frac{df}{dx}\frac{d^2x}{du^2}[/tex]

    humm...

    The first three equations was a attempt for show what would an integral of f (like an differential of f) and the implication shows the utility of the integral of f as an chain rule.

    If you get the last equation, ∫w, and multiply the equation by an arbitrary differential du, you'll have an chain rule of integrals in tree-dimensions:
    [tex]\int w du= \int f \frac{du}{dx}dx + \int f \frac{du}{dy}dy + \int f \frac{du}{dz}dz[/tex]
     
  5. Mar 31, 2014 #4

    Mark44

    Staff: Mentor

    So d2x would be d(dx). AFAIK, this doesn't mean anything. At least it's not anything I've ever seen. Also, as I mentioned earlier, you can't divide by dx, and you can't divide by d2x. The "fractions" dy/dx and d2y/dx2 are more notation than fractions that you can manipulate.

    Instead of merely manipulating symbols, as you seem to like to do, make up a function w = f(x, y, z), and see if your formula for ##\int wdu## has any relation to reality.
     
  6. Apr 4, 2014 #5
    hummm... so, tell me you, why no exist total and partial integral like in differentiation, that has partial and total differential. Why my analogy no make sense???
     
  7. Apr 4, 2014 #6

    Mark44

    Staff: Mentor

    Just off the top of my head, possibly it's because differentiation and integration aren't exactly inverse operations.
     
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