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From this example: How to calculate the pullback of a $k$-form explicitly

The answer said to think of $\alpha$ as a map, and I understand the working he is doing, however I am not sure how this conincides with my definition of a pullback. I was given:

For a smooth function $f:\mathbb{R^n} \to \mathbb{R^m}$ smooth with $p \in \mathbb{R^n}$ and $w$ a $k$-form on $\mathbb{R^n}$ we have that:

$f^\star w(p)((v_1)_p,...,(v_k)_p) = w(f(p))(Df(p)(v_1),...,Df(p)(v_k))$ now I can see why this definition makes sense, however I cannot see how I can use this definition to get the solution of say the example in the link given i.e. how can I use this definition to find $\alpha^\star w$ where $w = xy dx + 2z dy -y dz$ and $\alpha(u,v) = (uv,u^2,3u+v)$

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  • You wrote "The answer said to think of $\alpha$ as a map." But actually, the answer you linked to said NOT to think of $\alpha$ as a map, but instead as a substitution of variables. So in your example, just plug in $x=uv$, $y=u^2$, and $z=3u+v$, and expand. – Jack Lee Dec 14 '15 at 23:22
  • As $x=uv, \alpha^* dx= udv+vdu$, similarly $\alpha^dy= 2udu, \alpha^dz=3du+dv$, and you can finish the computation. – Thomas Dec 15 '15 at 13:03

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