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Let f: A → Rp be as in Theorem 5-1.

a. If x Є M = g-1(0), let h: U → Rn be the essentially unique diffeomorphism such that goh (y) = (y n – p + 1 . yn) and h (0) = x. Define f: Rn- p → f: Rn-p →Rn by f (a) = h (0, a). Show that is 1-1 so that the vectors f * ((e1)0). f* ((en-p)0) are linearly independent.

b. Show that the orientations μ can be defined consistently, so that M is orientable.

c. If P = 1, show that the components of the outward normal at are some multiple of D1g (x). . . Dng(x).

a. If x Є M = g-1(0), let h: U → Rn be the essentially unique diffeomorphism such that goh (y) = (y n – p + 1 . yn) and h (0) = x. Define f: Rn- p → f: Rn-p →Rn by f (a) = h (0, a). Show that is 1-1 so that the vectors f * ((e1)0). f* ((en-p)0) are linearly independent.

b. Show that the orientations μ can be defined consistently, so that M is orientable.

c. If P = 1, show that the components of the outward normal at are some multiple of D1g (x). . . Dng(x).

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