I'm coming from physics, but I figured this is a mathematical question, so here I am. I have a scalar $f(\vec{x},\vec{y})$ that is a function of two different points $\vec{x}$ and $\vec{y}$ in space. My problem arises from the fact that this function is evaluated at different points of the same trajectory. That is, I have a particle following a path parametrized by $\vec{z}(t)$ and my function looks like
\begin{equation} f\big[\vec{z}(t),\vec{z}(t')\big] \end{equation}
This is not a problem on itself. The problem is that the function gets evaluated on these different points of the trajectory $\vec{z}(t)$ after I take the derivative respect to the first entry of the function. So I get something like
\begin{equation} \frac{\partial f}{\partial x}(x,y) |_{\vec{x}=\vec{z}(t),\vec{y}=\vec{z}(t')} \end{equation}
The first question is: Is there a source in the literature where this type of thing shows up?
The second question is: How would you notate that derivative? derivative respect to $\vec{z}$ doesn't work because it looks like it's acting on the second entry as well. Is there some sort of notation for "Derive respect to whatever is in the first entry"?
$\endgroup$ 2 Reset to default
