By Dieudonne J.

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**Example text**

X n+1 ) = (x 1 )2 + · · · + (x n+1 )2 − r 2 . 1) for the ellipsoid with half-axes a 1 , . . , a n+1 . 7)). Show that it can be regarded as a hypersurface in Rn . 2 Hint: define g : Rn → R1 as g(A11 , A12 , . . 7). 4 Let M = {(x, y) ∈ R2 | y = f (x)}, where f : R → R. 10)). Draw for f (x) = tanh x. Hint: φ(x, y) = y − f (x); coordinate x. • It turns out that it is not possible to treat all manifolds by means of constraints (implicitly). 4)) that a manifold constructed by this method is necessarily orientable.

7). 4 Let M = {(x, y) ∈ R2 | y = f (x)}, where f : R → R. 10)). Draw for f (x) = tanh x. Hint: φ(x, y) = y − f (x); coordinate x. • It turns out that it is not possible to treat all manifolds by means of constraints (implicitly). 4)) that a manifold constructed by this method is necessarily orientable. There are, however, non-orientable manifolds, too. A more general approach is offered by a parametric expression of the latter. Within this scheme a manifold appears as the image of a smooth mapping f : A → Rn [x 1 , .

12 12 A comparison of the implicit and parametric ways of defining a manifold: in both cases mappings of Cartesian spaces Rm → Rn play an essential role. In the implicit way m ≥ n holds and the resulting manifold appears as the subset on the left (as the inverse image of (say) zero), in the parametric case m ≤ n and the manifold appears as the subset on the right (as the image of the map). e. the manifold M is a sphere S 1 (circle) (ii) the fact that the sphere S 1 appears on the right could be recognized (in advance) in the parameter space (on the left) as well.