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Let $M,N$ be two differentiable manifolds, $\phi:M\to N$ a diffeomorphism and $X$ a vector field on $M$. For example, one can determine the push forward $$(\phi_*X)(q)=\mathrm{D}\phi\left(\phi^{-1}(q)\right)\left(X(\phi^{-1}(q)\right),~q\in N$$ by calculating the differential $\mathrm{D}\phi$ in local coordinates. But in the case of embedded submanifolds of $\mathbb{R}^n$ I have seen (but am unable to reconstruct it) an alternative approach to calculate the differential: The basic idea is to identify the tangent spaces $T_pM$ and $T_qN$ with sub spaces of $\mathbb{R}^m$ respectively $\mathbb{R}^n$ via the inclusion map $i_M:M\to\mathbb{R}^m$: $$\mathrm{D}i_M(p)\left(T_pM\right)\subset T_{i_M(p)}\mathbb{R}^m\equiv\mathbb{R}^m.$$

I hope, someone knows how to calculate the differential $\mathrm{D}\phi$ with this approach.

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    $\begingroup$ I got lost during the second half of the question, where is $\Bbb R^3$ coming from? If you are mapping between two manifolds thought of as submanifolds of Euclidean space, then the ordinary derivative when thought of as a restriction will be fine, there is no need for charts. $\endgroup$ – muzzlator Apr 9 '13 at 6:31
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Writing the differential of $\phi':=i_N\circ\phi\circ i_M^{-1}:\mathbb{R}^m\to\mathbb{R}^n$ in local coordinates and identify $T_pM\subset T_{i_M(p)}\mathbb{R}^m\cong\mathbb{R}^m$ and $T_{\phi(p)}N\subset T_{i_N(\phi(p))}\mathbb{R}^n\cong\mathbb{R}^n$ $$\mathrm{D}\phi'(i_M(p))=\mathrm{D}\left(i_N\circ\phi\circ i_M^{-1}\right)(i_M(p)):\mathbb{R}^m\to\mathbb{R}^n$$ one can restrict $\mathrm{D}\phi'(p)$ to $T_pM\to T_{\phi(p)}N$ to obtain $\mathrm{D}\phi(p)$.

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