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I was trying to reverse engineer an equation from an article I was reading. The basic expressions I was working from were:

$ε = \frac{1}{2}w_x^2\tag{1}$

$b_1ε_tw_{xx} + b_2ε_{tt}w_{xx} + b_1ε_{xt}w_x + b_2ε_{xtt}w_x\tag{2}$

$(2)=2b_1w_xw_{xx}w_{xt} + 2b_2w_xw_{xx}w_{xtt} + b_1w_x^2w_{xxt} + b_2w_x^2w_{xxtt}\tag{3}$

I found that I was able to get their equation (3) but only by using derivations of the $ε$ terms using a specific method I'm not sure of.

Solely using the product rule applying all derivatives in a stepwise fashion we get the following set of equations we will call "Set A":


[SET A] All derivatives via stepwise product rule:

  • $ε = \frac{1}{2}w_x^2$
  • $ε_x = w_xw_{xx}$
  • $ε_t = w_xw_{xt}$
  • $ε_{xt} = w_{xt}w_{xx}+w_xw_{xxt}$
  • $ε_{tt} = w_{xt}^2 + w_xw_{xtt}$
  • $ε_{xtt} = w_{xtt}w_{xx} + 2w_{xt}w_{xxt} + w_xw_{xxtt}$

If you substitute "Set A" into (2) you get:

$2b_1w_xw_{xx}w_{xt} + b_2w_{xt}^2w_{xx} + 2b_2w_xw_{xx}w_{xtt} + b_1w_x^2w_{xxt} + 2b_2w_xw_{xt}w_{xxt} + b_2 w_x^2w_{xxtt}\tag{4}$

(4) at least superficially does not seem the same as (3).

If however, we assume that you can apply a double derivative of the same nature (ie. $y_{tt}$ or $y_{xx}$) in one step via the product rule so that $ε_{tt} = (\frac{1}{2}w_xw_x)_{tt} = w_xw_{xtt}$, we get "Set B" of definitions:


[SET B] Apply dual derivatives in one step:

  • $ε = \frac{1}{2}w_x^2$
  • $ε_x = w_xw_{xx}$
  • $ε_t = w_xw_{xt}$
  • $ε_{xt} = w_xw_{xxt} + w_{xt}w_{xx}$
  • $ε_{tt} = w_xw_{xtt}$ //DIFFERENT FROM SET A
  • $ε_{xtt} = w_{xx}w_{xtt}+w_xw_{xxtt}$ //DIFFERENT FROM SET A

If you substitute "Set B" into (2) you get (3) exactly, so these seem to be the derivatives they are using.

However this creates the impossible equality:

$ε_{tt} = w_xw_{xtt} = w_{xt}^2 + w_xw_{xtt}\tag{5}$

Is "Set B" valid? Is it a manifestation of the chain rule perhaps? ie. $y_{tt} = (y_t)_t$? What about "Set A"? I presume there is no way they can both be true.

I wonder if the authors could have made a mistake doing this. It's perhaps possible I made some other error, but I think it would be incredibly coincidental that using "Set B" derives their terms identically. Everything else in our equations line up except this part. So I do presume "Set B" is what they did.

I don't really understand the chain rule. I'm not that knowledgeable about derivatives. Which is mathematically valid: "Set A" or "Set B"?

Any clarification is appreciated. Thanks.

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  • $\begingroup$ The expressions don't appear to be equal $-$ yours has $9$ terms, whereas theirs has only $6$ (counting multiplicities). It looks like you made a mistake somewhere. $\endgroup$
    – TonyK
    Jan 19, 2021 at 17:38
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    $\begingroup$ In your expression for $\varepsilon_{xtt}$, your $w_{xx}w_{xxt}$ should be $w_{xt}w_{xxt}$. But that doesn't explain the discrepancy. I think perhaps your expression $(1)$ is wrong. $\endgroup$
    – TonyK
    Jan 19, 2021 at 17:45
  • $\begingroup$ Thanks TonyK. I was able to reverse engineer just by working backwards what I believe they used as the basic derivatives, but I'm not sure if they're valid. This is very peculiar to me. I'm wondering if you can take another look and let me know which of the two sets of potential basic derivatives are valid. Thanks. $\endgroup$
    – mike
    Jan 20, 2021 at 9:56
  • $\begingroup$ As I already pointed out, $(3)$ and $(4)$ can't possibly be equal, because $(3)$ has six terms and $(4)$ has nine terms (and all terms are positive). But $\varepsilon_{tt}$ is wrong in Set B. Perhaps the authors did make a mistake. $\endgroup$
    – TonyK
    Jan 20, 2021 at 10:16
  • $\begingroup$ By the way, you still haven't fixed the mistake in $\varepsilon_{xtt}$. $\endgroup$
    – TonyK
    Jan 20, 2021 at 10:21

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