Integral of x_2-x_1

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Note: the partial differential equation (on the third line) is quoted from Erpul, G., Gabriels, D., Norton, L.D., 2003. The Combined Effect of Wind and Rain on Interrill Erosion Processes. Lecture given at the College on Soil Physics Trieste, 3 -21 March 2003 (LNS0418015):174-182. [8 August 2012] and he makes a reference to "(Pedersen and Hasholt, 1995)" concerning the equation
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  • #2
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You need to be clear in your mind about the physical process intended by the 'Δ'.
You expression for ΔVx is fine, if it means the change in horizontal velocity of a raindrop as it descends from z1 to z2. But then you introduce the notion of a delta in time as though it were independent. For a given raindrop, a delta in vertical height will be accompanied by a delta in time.
To arrive at your final equation, you seem to have shifted from Δx = ∫Vx.dt to Δx = ∫ΔVx.dt, which doesn't make any physical sense to me.
  • #3
Thanks! I managed to solve the problem.
I think the point I got wrong is that when we have a FINITE integral, it already counts as a 'delta' or 'change', thus we don't need to find the 'integral of a delta'

On the other hand, when we have an INFINITE integral, then we may 'integrate deltas'. Then, Δx=∫ΔV.dt if we use infinite integrals.

A clear example would be using graphs. The finite integral already shows us the area BETWEEN two curves or lines, so we don't have to find a 'delta' because the finite integral already gives the 'delta'.
Using a delta in an infinite integral basically makes the infinite integral behave like a finite integral without a delta.
  • #4
I've written an explanation (the answer) here : [Broken]
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