We were learning about strain gauges and somehow we did this:
From
$R = \frac{\rho L}{A}$
to
$dR = \frac{\partial R}{\partial \rho}d\rho + \frac{\partial R}{\partial L}dl + \frac{\partial R}{\partial A}dA $
but I don't understand how this differential was formed?
Could anyone clarify this?
The next explanation is in "Physicist language". I bet Mathematicians to not read it to preserve their health (and by the way, to preserve my reputation).
Suppose you have a function of 3 variables : $f(x,y,z)$
If $x$ varies of a small value $dx$ while $y$ and $z$ are maintained constants, then $f$ varies of $df=\frac{\partial f}{\partial x}dx$
This has the same signifiance as $df=\frac{df}{dx}dx$ where $\frac{df}{dx}$ is the usual derivative. But one write $\frac{\partial f}{\partial x}$ instead of $\frac{df}{dx}$ to remember that only the variable $x$ varies.
$\frac{\partial f}{\partial x}$ is called "partial derivative" because it is the usual derivative, but with regard to only one variable when several variables are present, the other variables being constant.
Similarly :
If $y$ varies of a small value $dy$ while $x$ and $z$ are maintained constants, then $f$ varies of $df=\frac{\partial f}{\partial y}dy.$
If $z$ varies of a small value $dz$ while $x$ and $y$ are maintained constants, then $f$ varies of $dz=\frac{\partial f}{\partial z}dz.$
Now, if $x$ , $y$ and $z$ varies together of respectively $dx$ , $dy$ , $dz$ , the total variation of $f$ is the sum of the three above variations : $$df=\frac{\partial f}{\partial x}dx+\frac{\partial f}{\partial y}dy+\frac{\partial f}{\partial z}dz$$
In your "resistance formula" change the notations : Replace $R$ by $f$ , replace $\rho$ by $x$, replace $L$ by $y$ and replace $A$ by $z$ and you have the explanation.
I beg pardon to Mathematicians who did read it despite my warning and if they don't agree with Nonstandard Analysis.