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demonelite123
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my book defines an orientation preserving parametrization of a manifold as one such that:
Ω(D1γ(u), ..., Dkγ(u)) = +1 for all u in the domain of γ, where D1,...Dk are the derivatives of the parametrization γ.
my book also defines the orientation of a surface in R^3 by Ω(v1,v2) = sgn det[n(x), v1, v2] where sgn takes the sign off a real number and n(x) is a transverse vector field and v1 and v2 form a basis for the tangent space of the surface at the point x. if we choose n(x) to be D1γ X D2γ (the cross product) then Ω(D1γ, D2γ) = sgn det[D1γ X D2γ, D1γ, D2γ] and it is simple to show that this determinant is > 0 for all input values. then by the first definition stated above, if you choose n(x) = D1γ X D2γ, then γ must preserve the orientation of the surface in R^3.
however my book seems to contradict itself when it gives an example. let M be the torus obtained by choosing R>r>0 and taking the circle of radius r in the (x,z) plane that is centered at x = R and z = 0 rotating it around the z axis. assume it is orientated by the outward pointing normal. does γ(u,v) = ((R+rcos u)cos v, (R+rcos u) sinv, rsin u)) preserve the orientation?
my book went on to calculate D1γ and D2γ as well as D1γ X D2γ and letting n(x) = D1γ X D2γ. my book presented a proposition right before this example that said if γ preserves orientation at a single point in U (domain of γ) then it preserves orientation at every point of U. the book calculated n = D1γ X D2γ at the point γ(0,0) and got n = -r(r+R)[1, 0, 0] (column vector). which it then says points inwards and thus does not preserve orientation. i am confused because earlier the book said that if you chose n = D1γ X D2γ then it will preserve orientation. any help is clearing up this matter is greatly appreciated.
Ω(D1γ(u), ..., Dkγ(u)) = +1 for all u in the domain of γ, where D1,...Dk are the derivatives of the parametrization γ.
my book also defines the orientation of a surface in R^3 by Ω(v1,v2) = sgn det[n(x), v1, v2] where sgn takes the sign off a real number and n(x) is a transverse vector field and v1 and v2 form a basis for the tangent space of the surface at the point x. if we choose n(x) to be D1γ X D2γ (the cross product) then Ω(D1γ, D2γ) = sgn det[D1γ X D2γ, D1γ, D2γ] and it is simple to show that this determinant is > 0 for all input values. then by the first definition stated above, if you choose n(x) = D1γ X D2γ, then γ must preserve the orientation of the surface in R^3.
however my book seems to contradict itself when it gives an example. let M be the torus obtained by choosing R>r>0 and taking the circle of radius r in the (x,z) plane that is centered at x = R and z = 0 rotating it around the z axis. assume it is orientated by the outward pointing normal. does γ(u,v) = ((R+rcos u)cos v, (R+rcos u) sinv, rsin u)) preserve the orientation?
my book went on to calculate D1γ and D2γ as well as D1γ X D2γ and letting n(x) = D1γ X D2γ. my book presented a proposition right before this example that said if γ preserves orientation at a single point in U (domain of γ) then it preserves orientation at every point of U. the book calculated n = D1γ X D2γ at the point γ(0,0) and got n = -r(r+R)[1, 0, 0] (column vector). which it then says points inwards and thus does not preserve orientation. i am confused because earlier the book said that if you chose n = D1γ X D2γ then it will preserve orientation. any help is clearing up this matter is greatly appreciated.