Sometimes in physics they do things like this one:
If $dq=f\left(x\right)\cdot dr$ then $\frac{dq}{dt}=f\left(x\right)\cdot \frac{dr}{dt}$ Which mathematically is a wrong deduction. Is there any way to justify the required steps to make it a rigorous deduction? -for example using the inverse function theorem or so-
Thanks a lot pals, Karan
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If the first expression is being interpreted as a statement about differentials, I see nothing wrong with what's written. The differential $dq$ equals $\frac{dq}{dt}dt$ while $dr$ equals $\frac{dr}{dt}dt$. Making these substitutions gives $$\frac{dq}{dt}dt = f(x)\frac{dr}{dt}dt$$ and comparing coefficients gives $\frac{dq}{dt} = f(x)\frac{dr}{dt}$ as claimed.
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An example : let $f$ be a smooth function in $x,y,z,t$, then $$ df = \frac{\partial f}{\partial t} dt + \frac{\partial f}{\partial x} dx+\frac{\partial f}{\partial y} dy +\frac{\partial f}{\partial z} dz$$
Now if $x, y ,z$ are implicit functions in $t$, you have $$ dx = \frac{\partial x}{\partial t} dt \,; dy = \frac{\partial y}{\partial t} dt ; dz = \frac{\partial z}{\partial t} dt$$
Which gives you $$ df = \left(\frac{\partial f}{\partial t} + \frac{\partial f}{\partial x} \frac{\partial x}{\partial t} + \frac{\partial f}{\partial y} \frac{\partial y}{\partial t} + \frac{\partial f}{\partial z} \frac{\partial z}{\partial t} \right)dt$$
Then $\frac{df}{dt}$ is just a notation to say $\frac{\partial f}{\partial t} + \langle\vec \nabla f | \vec v\rangle$.
Hence, what is done in Physics is perfectly rigorous if you understand their implicit notations.
I would say the first is an abuse of notation, while the second is well defined if the variables have the right properties. You can make it rigorous by expanding everything in Taylor series. The first one would be seen as a statement about the first order terms in the Taylor series and a claim that the second order terms become negligible in the limit $\Delta t \to 0$. The second takes advantage of all that. Your functions have to be "nice" for all this to work, differentiable of course, but probably uniformly continuous or Lipschitz. Physics functions tend to be so because if the derivatives get too big that represents infinite energy or force or something.