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I am trying to solve the following question

Let $\mathbf r(u,v)$ be a parametrised smooth surface in $\Bbb R^3$ with $(u,v) \in U$, a connected open subset of $\Bbb R^2$. Let $\mathbf n: U \to S^2$ be the Gauss map that assigns to each point $\mathbf r(u,v)$ of the surface its unit normal. Suppose that $\mathbf n$ is bijective onto its image and that the Gaussian curvature K is nowhere zero in U. Show that the area of $\mathbf n(U)$ is equal to the absolute value of $\int K dA$

I am not really sure where to start, so any hint would be appreciated.

Orlly
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    Hey! What's your background on the topic? :) Do you know how to represent the derivative $dn$ of $n$ as a map between tangent spaces? And what definition do you have for the Gaussian curvature? – Azur Nov 17 '20 at 01:02
  • Hi! I am studying differential geometry at an undergraduate level, so my background is quite limited. The definition of Gaussian curvature that I'm using is the ratio (LN-M^2)/(EG-F^2), where these are the coefficients in the first and second fundamental forms of a surface. I'm not too familiar with tangent spaces overall – Orlly Nov 17 '20 at 01:09
  • Alright, I'm not too familiar with that side of the topic so I'll leave that for someone more knowledgeable :D – Azur Nov 17 '20 at 01:10
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    I'm betting that your text or instructor gave you a bit more on this. But if not, see the Remark on p. 51 of my (free) differential geometry text, linked in my profile. – Ted Shifrin Nov 17 '20 at 01:44

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In case you want an explicit computation. Recall $$dA=\mid r_u \wedge r_v\mid dudv= ((r_u \wedge r_v) \cdot(r_u \wedge r_v))^{1/2}dudv=\sqrt{EG-F^2}dudv$$ by the scalar quadruple product. Then: $$Area(n(U))=\int\int \mid n_{u} \wedge n_{v}\mid dudv$$$$=\int\int \mid\frac{LN-M^2}{ \sqrt{EG-F^2}} n\mid du dv$$$$=\int\int \sqrt{EG-F^2}\mid K \mid du dv=\int\int\mid K\mid dA$$ (That $n_{u} \wedge n_{v}=\frac{LN-M^2}{ \sqrt{EG-F^2}} n$ can be found on page 65 here: https://courses.maths.ox.ac.uk/node/view_material/48840 and is a result of some simple identities and the fact that a unit vector is orthogonal to its derivatives)

HappyChappy28
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