I'm confident in my math ability, but how is it that by using the chain(adsbygoogle = window.adsbygoogle || []).push({});

rule...

[tex]

W_{x_1 \rightarrow x_2} = \int^{x_2}_{x_1} m \frac{dv}{dt} dx

[/tex]

can be turned into

[tex]

W_{x_1 \rightarrow x_2} = \int^{x_2}_{x_1} m \frac{dv}{dx} \frac{dx}{dt} dx = \int^{v_2}_{v_1}mv dv

[/tex]

?

I understand the concept of using chain rule to make velocity depend on position which is dependent on time

[tex]v(t)=v(x(t))[/tex]

[tex]v'(t)=v'(x(t))x'(t)[/tex]

where

[tex]\frac{dv}{dt}=\frac{dv}{dx}\frac{dx}{dt}[/tex]

however even with the above in mind integration by substitution is defined as:

[tex]\int^{b}_{a} f(g(t))g'(t)dt = \int^{g(b)}_{g(a)} f(x)dx [/tex]

which means the integral "dx" should be a "dt" instead since time is the base independent variable.

[tex] \int^{x_2}_{x_1} m v'(x(t))x'(t) dt [/tex]

and after substitution should be in the form

[tex] \int^{x(x_2)}_{x(x_1)} mv'(x)dx [/tex]

but obviously that makes no sense either since the limits of integration are in terms of "x" and x(t) needs to have inputs of time. where did i go wrong?

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# Confusion on chain rule substitution

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