I this it should apply $$dx=x(X+dX)-x(X)$$
But I am not sure about the answer. Is it$$\frac{f(R+dR{\mathbf{i_r}},t)}{R+dR\mathbf{i_r}}(R\mathbf{i_r}+dR\mathbf{i_r})-\frac{f(R,t)}{R}R\mathbf{i_r}$$?
I tried to do grad(x) = $$grad(\frac {f(R,t)}{R} X)$$
and by product rules:
=$$\frac{f(R,t)}{R} ∇X + X ⋅ ∇(\frac {f(R,t)}{R})$$
If I am correct, ∇X should be I, so
=$$\frac{f(R,t)}{R} I + X ⋅ ∇(\frac {f(R,t)}{R})$$
This is by far what I have done.
Question is extracted from "Ellad B Tadmor, Ronald E Miller, Ryan S Elliott - Continuum mechanics and thermodynamics From fundamental concepts to governing equations".
I just got stuck at part (a). I think if part(a) is solved, I may be able to do the other parts.
Thanks for your answers.
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