Can someone please explain how the following
$$ \textbf{F} \cdot d\textbf{s} = -dV$$
is equivalent to
$$ F_s = -\frac{\partial V}{\partial s}$$
using some intermediate steps. I don't follow this in Goldstein's text. Thanks.
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$$\text d f(x)=f'(x)\text d x\equiv\frac{\partial f}{\partial x}\text d x.$$
The second equation you posted is of course only a component representation, the vectors in the first equation are most definately higher dimensional (3 dimensional). And so the derivative in the second equation is chosen such that it goes in the direction of the path, denoted y s. I'm afraid $F_s$ is supposed to denote the "s-component".
answered Jul 20, 2012 at 11:50
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$\begingroup$ Thanks, I'm beginning to get it. But why does $ds$ become a partial derivative in the second equation? $\endgroup$– JoebevoJul 20, 2012 at 12:11
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$\begingroup$ @Joebevo: It doesn't. As I said, the second equation is only the component representation of the first one: The expression $V(\vec x)$ gets evaluated to a scalar at every point $\vec x$ and $\text d V$ is considere to be the change of $V$ in the direction of the path $s$. So by the equation I posted you have $\text d V = V'(s)\text d s\equiv \frac{\partial V}{\partial s}\text d s$. Plug that into the first equation and you get the second equation - apart from the $\text d s$ (what's left is "only the component expression") and a complication that they dropped the vector notation for $F$. $\endgroup$ Jul 20, 2012 at 12:34
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$\begingroup$ Shouldn't the equation you posted be $\text d f(x)=f'(x)\text d x\equiv\frac{df}{dx}\text d x$, since f is a function only of x? According to my knowledge, the partial derivative comes in when the function is one of multiple variables. $\endgroup$– JoebevoJul 21, 2012 at 7:54
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$\begingroup$ @Joebevo: In the link you see the definition of the exterior derivative and that it's defined with a partial one. That's the save way. But yes, it's only necessary if there are more than one variable. If there is only one variable then there is no difference. $\endgroup$ Jul 21, 2012 at 12:17