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What are generic terms for integration/summation parameters?

  1. Oct 22, 2015 #1
    This is not only a question strictly about mathematics, but in science or any other quantitative field in which there is an integration - or a summation that is like a discrete integration.

    [ A ] the parameter that is considered the input variable for the integration/summati - i.e., the x of dx

    [ B ] the parameter that is the function being integrated/summed - f( x )

    [ C ] the integration/summation result - { ∫ f( x ) dx } OR { Σ [ f( x ) Δx ] }

    Obviously the fact that one is a summation while the other is integration (which itself is simply the limit of the summation being an infinite number of discrete quantities) should make no difference in the abstract terminology. Also, speaking only for physics here, there are a number of different such summation relationships ...

    Mechanical Energy = Σ [ Force(Distance) * Δ Distance ]

    Momentum = Σ [ Force(Time) * Δ Time ]

    Elastic Energy = Σ [ Stiffness(Deformation) * Δ Deformation ] = Σ [ Elasticity(Strain) * Δ Strain ]

    Thermomechanical Energy = Σ [ Pressure(Volume) * Δ Volume ]

    Heat = Σ [ Temperature(Entropy) * Δ Entropy

    Energy = Σ [ Power( Time ) * Δ Time ]

    Voltage = Σ [ ElectricalField( Distance ) * Δ Distance ]

    ElectricalCharge = Σ [ Current(Time) * Δ Time ]

    Flux = Σ [ Field( 2-D coordinates on surface ) * Δ Area ]

    ... to name a few. Likewise, the function here is the derivative of the integration result with respect to the differential parameter. It seems that there must be a nice set of elegant terms to describe any such relationship; the best I can come up are ...

    [ A ] the displacement function

    [ B ] the forcing function

    [ C ] the accumulation

    ... which obviously is inconsistent. I think the term for [ C ] sounds pretty good, but I can't come up with a term that is as generic as "accumulation" to describe [ A } & [ B ]. Surely some great commentator on mathematics has come up with such a set of nice terms.
  2. jcsd
  3. Oct 22, 2015 #2
    In mathematics and science... isn't it just?

    A. The independent variable
    B. The dependent variable
    C. Instantaneous change in the dependent variable

    For instance, the integration of velocity dependent on time is acceleration. Time is clearly the independent variable upon which velocity occurs and the instantaneous change in velocity is known as acceleration?
  4. Oct 22, 2015 #3


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    So far as I know, only B has an official term, which is integrand and summand respectively.
  5. Oct 23, 2015 #4


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    The Wiki article on integration uses some reasonable terms. "differential" for the dx, "variable of integration" for the x and a choice of "integral", "definite integral", "indefinite integral" or "antiderivative" depending on how you want to consider the result.

    It uses "integrand" for the function being integrated as Andrewkirk has already indicated.

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