- #1
milesyoung
- 818
- 67
Hi, I'm new to these forums so not exactly sure where to place this question, although calculus seems a good bet, so here goes:
I'm currently taking a mechanics course at my university (current subject is work/energy), and I'll just post this snippit from our textbook (Physics for Scientists and Engineers with Modern Physics 7th edition, Jewett/Serway):
[snippit]
[tex]
\begin{equation*}
W_{net}=\int^{x_f}_{x_i}\sum F\,dx
\end{equation*}
[/tex]
Using Newton's second law, we substitute for the magnitude of the net force [itex]\sum F=ma[/itex] and then perform the following chain-rule manipulations on the integrand:
[tex]
\begin{align*}
W_{net}&=\int^{x_f}_{x_i}ma\,dx=\int^{x_f}_{x_i}m\frac{dv}{dt}\,dx=\int^{x_f}_{x_i}m\frac{dv}{dx}\frac{dx}{dt}\,dx=\int^{v_f}_{v_i}mv\,dv\\
W_{net}&=\frac{1}{2}{mv_f}^2-\frac{1}{2}{mv_i}^2
\end{align*}
[/tex]
[/snippit]
How is:
[tex]\frac{dv}{dt}=\frac{dv}{dx}\frac{dx}{dt}[/tex]
legal in this context?
I know the Chain Rule states the following:
[tex]
\begin{equation*}
\frac{dy}{dx}=\frac{dy}{du}\cdot\frac{du}{dx}
\end{equation*}
[/tex]
But this is valid only (to my understanding) for the derivative of the composite of two functions.
If anyone could help me sort this out, I'd be much obliged.
Thanks.
I'm currently taking a mechanics course at my university (current subject is work/energy), and I'll just post this snippit from our textbook (Physics for Scientists and Engineers with Modern Physics 7th edition, Jewett/Serway):
[snippit]
[tex]
\begin{equation*}
W_{net}=\int^{x_f}_{x_i}\sum F\,dx
\end{equation*}
[/tex]
Using Newton's second law, we substitute for the magnitude of the net force [itex]\sum F=ma[/itex] and then perform the following chain-rule manipulations on the integrand:
[tex]
\begin{align*}
W_{net}&=\int^{x_f}_{x_i}ma\,dx=\int^{x_f}_{x_i}m\frac{dv}{dt}\,dx=\int^{x_f}_{x_i}m\frac{dv}{dx}\frac{dx}{dt}\,dx=\int^{v_f}_{v_i}mv\,dv\\
W_{net}&=\frac{1}{2}{mv_f}^2-\frac{1}{2}{mv_i}^2
\end{align*}
[/tex]
[/snippit]
How is:
[tex]\frac{dv}{dt}=\frac{dv}{dx}\frac{dx}{dt}[/tex]
legal in this context?
I know the Chain Rule states the following:
[tex]
\begin{equation*}
\frac{dy}{dx}=\frac{dy}{du}\cdot\frac{du}{dx}
\end{equation*}
[/tex]
But this is valid only (to my understanding) for the derivative of the composite of two functions.
If anyone could help me sort this out, I'd be much obliged.
Thanks.