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Let $f:\mathbb{R}^2 \to \mathbb{R}^2$ be a function mapping $$(x, y) \to \left(e^{2x}, xy\right).$$

How do you compute pulling back form of a 2-form $$\alpha(x, y)=xy(dxdy),$$ in other words, $f^*(\alpha)$?

gt6989b
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jfcjohn
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    I suspect you want the wedge product between $dx$ and $dy$. Also, where exactly is the problem, since this looks very straight-forward? I assume you are taking some kind of class and have a textbook. Are you getting stuck at some particular point? – Lukas Geyer Sep 24 '15 at 18:59
  • I was asked to compute pulling back form of multiple-forms including 3, 4... I am just generally confused of how to approach problems of this kind. I do have notes of a formula which makes use of a function g in target space of f, but without process of proof it is very hard for me to understand how/why it works. There is no textbook, and I've been googling but can't find a strategy of solving such problems. This is my situation, any suggestions? – jfcjohn Sep 24 '15 at 20:20

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Pullback commutes with the wedge product, and $xy \, dx \, dy$ is the wedge product of $x, y, dx$ and $dy$ (two $0$-forms and two $1$-forms respectively). So it suffices to compute the pullback of each of these.

The pullback of $x$ is the $x$-component of your function, and similarly for $y$. So $f^{\ast}(x) = e^{2x}$ and $f^{\ast}(y) = xy$.

Pullback also commutes with the exterior derivative, so the pullback of $dx$ is the exterior derivative of the pullback of $x$, and similarly for $y$. So

$$f^{\ast}(dx) = d e^{2x} = 2 e^{2x} dx$$ $$f^{\ast}(dy) = d (xy) = x \, dy + y \, dx.$$

So in total we get

$$f^{\ast}(xy \, dx \, dy) = e^{2x} (xy)(2e^{2x} \, dx)(x \, dy + y \, dx) = 2x^2 y e^{4x} \, dx \, dy.$$

Qiaochu Yuan
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  • Thanks for you reply! I noticed in your wording that there should be a precondition for equating pullback of original product (xydxdy above) and each of the component (x, y, dx, dy respectively). Is it because number of for each form in product is the same? Also, is there any recommended textbook (hopefully) about pullbacks I can refer to? – jfcjohn Sep 24 '15 at 21:33
  • @szerzp: I don't understand the question. In any case, any textbook on differential geometry should include a chapter on differential forms and tell you what sorts of things are natural in the sense that they respect pullback. – Qiaochu Yuan Sep 24 '15 at 22:47