Let \(\alpha,\beta,\xi\) be smooth \(1\)-forms on \(\mathcal{X}\) and \(f,h \in C^{\infty}( \mathcal{X}, \mathbb{R})\) functions. Verify the following:

1. \(\alpha\wedge\beta=-\beta\wedge\alpha\) so that \(\alpha\wedge\alpha=0.\)

2. \((\alpha+\beta)\wedge \xi=\alpha\wedge\xi+\beta\wedge\xi.\)

3. \((f\alpha)\wedge\beta=\alpha\wedge(f\beta)=f(\alpha\wedge\beta).\)

Now use these properties to show that \[du\wedge dv=r dr\wedge d\theta,\] where \((r,\theta)\) denote the polar coordinates on \(\mathbb{R}^2\) and we set \((u,v)=(r\operatorname{cos}(\theta),r\operatorname{sin}(\theta)).\)

Let \(M_1,M_2,M_3\subset\mathbb{R}^3\) be embedded surfaces. Prove the following:

1. If \(\Phi\colon M_1\rightarrow M_2\) is an isometry, then \(\Phi^{-1}\colon M_2\rightarrow M_1\) is an isometry.

2. If \(\Phi\colon M_1\rightarrow M_2\) and \(\Psi\colon M_2\rightarrow M_3\) are isometries, then \(\Psi\circ\Phi\colon M_1\rightarrow M_3\) is an isometry.

3. The collection of isometries of \(M_1\) form a subgroup of the diffeomorphism group of \(M_1.\) (Note: This is called the isometry group of \(M_1\)).

Let \(M_1,M_2\subset\mathbb{R}^3\) be embedded surfaces. Prove that the following hold:

1. A diffeomorphism \(\Phi\colon M_1\rightarrow M_2\) is an isometry if and only if the length of any parametrized curve \(\gamma\colon [a,b]\rightarrow M_1\) is equal to the length of the image curve \(\Phi\circ \gamma\colon [a,b]\rightarrow M_2.\)

2. An isometry \(\Phi\) is distance-preserving. That is, if \(p,q\in M_1\) and \(\gamma\colon[a,b]\rightarrow M_1\) is a curve such that \(\gamma(a)=p\) and \(\gamma(b)=q,\) then \(\gamma\) is distance-minimizing if and only if the curve \(\Phi\circ\gamma\colon[a,b]\rightarrow M_2\) is distance minimizing.