High School Mathematics Contest

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## 2014 Contest Logo

 We owe the T-shirt design and our 2014 contest logo design to Prof. Gestur Olafsson of LSU Math Department. We also thank Richard Frnka who created the electronic files.

The sphere $$\mathrm{S}^n=\{\mathbf{v}\in \mathbb{R}^{n+1}\mid \|\mathbf{v}\|=1\}$$ is a closed compact manifold in $$\mathbb{R}^{n+1}$$. It is well know that polar-coordinates play an important role in analysis on $$\mathbb{R}^{n+1}$$, in particular, in solutions of differential equations with rotational symmetry. One can also define polar coordinates on the sphere. For that we embed $$\mathrm{S}^{n}$$ into $$\mathrm{S}^{n+1}$$ by $$\mathbf{u}\mapsto (0,\mathbf{u})$$. If $$\mathbf{v}\in \mathrm{S}^n$$ then we write $$\mathbf{v}=(x_1,\mathbf{x})$$ with $$\mathbf{x}\in \mathbb{R}^n$$. As $$\|\mathbf{v}\|=1$$ it follows that $$x_1^2+\|\mathbf{x}\|^2=1$$. We can therefore write $$x_1=\cos (t)$$ and $$\|\mathbf{x}\|=\sin (t)$$ for some $$t\in [0,\pi]$$. If $$\mathbf{x}\ne\mathbf{o}$$ then $$\mathbf{u}=(1/\sin t)\, \mathbf{x}\in \mathrm{S}^{n-1}$$ and $$\mathbf{v}=\cos (t)\mathbf{e}_1+\sin (t) \mathbf{u}$$ where $$\mathbf{e}_1=(1,0,\ldots ,0)^t$$ is the first standard basic vector for $$\mathrm{R}^{n+1}$$. The coordinates $$(t,\mathbf{u})\in (0,\pi)\times \mathrm{S}^{n-1}$$ are the polar-coordinates for the sphere.  The sphere is an example of a homogeneous manifold. Denote by $$\mathrm{GL} (n+1,\mathbb{R})$$ the group of all invertible $$(n+1)$$-matrices. The special orthogonal group $$\mathrm{SO} (n+1)$$ is defined by $\mathrm{SO} (n+1)=\{A\in \mathrm{GL} (n+1,\mathbb{R})\mid (\forall \mathbf{v},\mathbf{w}\in\mathbb{R}^{n+1})\, \langle A\,\mathbf{v} ,A\,\mathbf{w} \rangle =\langle \mathbf{v} , \mathbf{w}\rangle \text{ and } \det A=1\}\, .$ If $$\mathbf{v}\in \mathrm{S}^n$$ then we also have $$\|A\,\mathbf{v}\|=1$$ and hence $$A\, \mathbf{v}\in \mathrm{S}^n$$. Write $$A=[\mathbf{a}_1,\ldots , \mathbf{a}_{n+1}]$$ where $$\mathbf{a}_1,\ldots ,\mathbf{a}_{n+1}$$ are column vectors. We have $$A\,\mathbf{e}_1=\mathbf{a}_1$$. On the other hand, if $$\mathbf{v}\in\mathrm{S}^{n}$$ then we can extend $$\mathbf{a}_1=\mathbf{v}$$ to a positively oriented orthonormal basis $$\mathbf{a}_1,\ldots ,\mathbf{a}_{n+1}$$ of $$\mathbb{R}^{n+1}$$. Then $$A=[\mathbf{a}_1,\ldots ,\mathbf{a}_{n+1}]\in\mathrm{SO} (n+1)$$ and $$A\,\mathbf{e}_1=\mathbf{v}$$. The group that fixes the ''north pole'' $$\mathbf{e}_1$$ is the group $$\mathrm{SO} (n)$$ and hence $$\mathrm{S}^{n}\simeq \mathrm{SO} (n+1)/\mathrm{SO} (n)$$ where the isomorphism is given by $$A/\mathrm{SO} (n)\mapsto A\mathbf{e}_1$$.

 Contest organizer: Contest e-mail:   Contest web-page: Mark Davidson, phone: (225) 578-1581 contest@math.lsu.edu www.math.lsu.edu/~contest