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B Derivation of exact differential

  1. Jul 29, 2017 #1
    Exact differential of a scalar function f takes the form of

    ∇f⋅dr=Σ∂ifdxi (where dr is a vector)
    f:R->Rn


    and I am not sure why this equation is valid in the sense that if we integrate the equation,

    ∫∇f⋅dr=∫{Σ∂ifdxi}
    ∫df=∫{Σ∂ifdxi}

    the above equation is true because integration is a linear operator, and if we think of R.H.S only,

    ∫{Σ∂1fdx1}+∫{Σ∂i≠1fdxi≠1}
    =f+c(x2,x3...,xn)+∫{Σ∂i≠1fdxi≠1}

    and if we think of each integration of differential w.r.t variable x1,x2,x3 and so on, it should generate f+c respectively as the result of each integration.

    therefore R.H.S has to be

    nf+C(x1,x2,x3....,xn) but according to textbook, it says we dont add up each integration but we compare them to eradicate constants and "merge" each equation to one right answer, f.

    I am a little bit confused about how to interpret the integration within the quotation mark ∫df="∫"{Σ∂ifdxi} because it seems it is linearly applied to each of partial differential, but does not spit out nf(n number of partial differentials so there must be n number of integrations on them so adding them up would give nf+C(x1,x2,x3....,xn)

    Please enlighten me. I would really appreciate
     
  2. jcsd
  3. Jul 31, 2017 #2
    As you noted, the gradient operator takes a scalar in R and gives a vector in $$R^n$$ and each derivative of f in the sum gives a component of the resulting vector. The key here is that the differential operators all act on f and they act differently. For instance you can have f = xy, taking the 2D gradient gives df/dx = y and df/dy = x. The scalar function f that produces the derivative is not the sum of derivatives which is 2*xy but is instead f = xy. This is what the book means by compare, eradicate and merge. You know there must be an xy term from integrating the x derivative and similarly for the y derivative but the result is not their sum in this case, you can test this by differentiating it again to see if it gives the same gradient.
     
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