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I have a line $L\subset\mathbb{C}^n$ which is parametrized by $x_1=a_1t, x_2=a_2t,\dots, x_n=a_nt$, a function $f(x_1,\dots,x_n)$, and I want to look at the restriction of $f$ onto $L$. This is just $f(a_1t,\dots,a_nt)$, but the part I'm having trouble with is that I want an expression for $\partial f/\partial x_i$ restricted to L, i.e. as a function of $t$.

Is there some simple way to do this with the chain rule? Of course simply taking $\partial f/\partial x_i = dt/dx_i \ df/dt$ doesn't work. Is there any simple expression for the derivative that does not directly involve $\partial f/\partial x_i$?

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up vote 2 down vote accepted

When you sit in a space ship with a thermometer in your hand you can measure the felt temperature $t\to u(t):=f\bigl({\bf x}(t)\bigr)$, and you can calculate its time derivative ${du\over dt}$. When you know your exact flight-path $t\mapsto{\bf x}(t)$ you will be able to say something about the directional derivative of the underlying temperature function $(x_1,x_2,x_3)\mapsto f(x_1,x_2,x_3)$ in your momentaneous flight direction $\dot{\bf x}(t)$, but there is no way to get information about the temperature change in other directions, in particular in the direction of the coordinate axes. Therefore from ${du\over dt}$ alone you will not get information about the partial derivatives ${\partial f\over\partial x_i}$. A more down to earth argument would be the following: From one number ${du\over dt}$ you cannot distill three numbers ${\partial f\over\partial x_i}$ if no extra dependencies are present in the given situation.

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