My textbook (nonlinear dynamics and chaos, strogatz), uses Green's theorem at some point (the actual use is not relevant for my question I hope):

given a real valued function $g(\vec x)$ such that $\nabla\cdot (g\dot x)$ [...] then using Green's theorem, $$\int\int_A\nabla\cdot(g\dot {\vec x})dA=\int_Cg\dot {\vec x }\cdot n dl $$ where

What does $g\dot {\vec x}$ mean here? $g$ is a function, so why isn't it written $g(\dot {\vec x})$? There must be a difference between the meaning of these two notations.

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    $\begingroup$ It may mean $g\cdot \frac{dx}{dt}.$ Physicists use dots for derivatives sometimes. $\endgroup$ – Sean Roberson Sep 25 '16 at 16:03
  • $\begingroup$ I know what the $\dot x$ means. my confusion is about $g$ $\endgroup$ – user56834 Sep 25 '16 at 16:44
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    $\begingroup$ You're just taking the argument of div to be $g$ times $\dot x$. $\endgroup$ – Sean Roberson Sep 25 '16 at 16:45
  • $\begingroup$ So you're saying $g\dot {\vec x}$ means $g(\vec x)*\dot {\vec x}$? $\endgroup$ – user56834 Sep 25 '16 at 19:42

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