Author Topic: Substitution in integral  (Read 1442 times)

Jiacheng Ge

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Substitution in integral
« on: September 10, 2018, 09:49:24 PM »
dt means with respect to t.

dw means with respect to w.

w’(t)dt means take derivative to w with respect to t.

How can we substitute w’(t)dt with dw?

dw is a prepositional phrase but w’(t)dt is a complete action.

« Last Edit: September 17, 2018, 05:59:57 PM by Victor Ivrii »

Nan Choi

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Re: substitution in integral
« Reply #1 on: September 10, 2018, 10:53:34 PM »
This is a really basic way to look at it:

Since $w'(t)$ means the derivative of $w$ with respect to $t$, we can write:

$w'(t) = \frac{dw}{dt}$

So, multiplying both sides by $dt$ gives:

$w'(t)dt = dw$

Jiacheng Ge

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Re: substitution in integral
« Reply #2 on: September 11, 2018, 03:13:09 PM »
dt cannot exist and appear alone. How can we multiply dt on both sides since dt is not a number?

Tzu-Ching Yen

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Re: substitution in integral
« Reply #3 on: September 12, 2018, 06:08:56 AM »
dt can exist alone. Look up the technique called separation of variable.

Victor Ivrii

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Re: substitution in integral
« Reply #4 on: September 12, 2018, 06:35:01 AM »
dt cannot exist and appear alone. How can we multiply dt on both sides since dt is not a number?
Please reread "Differential" in Calculus I

Jiacheng Ge

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Re: substitution in integral
« Reply #5 on: September 12, 2018, 11:38:24 PM »
After reading it, I have a follow-up question. What's the difference between  w’(t)dt,  dw,  dw/dt,  dw(t)/dt?

Victor Ivrii

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Re: substitution in integral
« Reply #6 on: September 13, 2018, 01:23:02 AM »
After reading it, I have a follow-up question. What's the difference between  w’(t)dt,  dw,  dw/dt,  dw(t)/dt?
$w’(t)dt=dw$ and $dw/dt=dw(t)/dt$. I am afraid, with questions like this,  you need to retake Calculus I.
« Last Edit: September 15, 2018, 04:20:11 PM by Victor Ivrii »

Omar Kidwai

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Re: substitution in integral
« Reply #7 on: September 17, 2018, 05:18:54 PM »
Victor, I would note that such a thing is not rigorously defined, at least not in Calculus I...

I like to think of $dt$ and $dw$ written alone (as opposed to, e.g. $dw/dt$) as semi-meaningless symbols which you manipulate as a mnemonic for actual meaningful operations. You can do everything without them if you like.
« Last Edit: September 17, 2018, 06:00:24 PM by Victor Ivrii »

Victor Ivrii

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Re: substitution in integral
« Reply #8 on: September 17, 2018, 05:59:46 PM »
Omar, indeed, one can do without $dw$ etc but it would require too much efforts. Too much pain with a little gain.