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Can you name the Differential Geometry Answers?

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QuestionAnswerExtra Info.
3: f:U->R smooth and gradf(p)=0 then p is called what?
3: What is given by the formula -(a'xa'').a'''/||a'xa''||^2
3: U,V open sets, then a smooth map h:V->U is called what if it is bijective and the inverse is smooth?
1: Integral from u0 to u of ||a'(u)||dt is the
1: The image a(I) of an open interval I also known as this
3:On surface: x a smooth map, x a homomorphism, partial derivatives w.r.t u and v of x are lin. ind. then what do we call x and x^-1 (give soln as x and x^-1)
2: Name of the point of a curve a where k'(u)=0
3: Torsion defined by this formula (let torsion = T) (give both sides of equation)
1: The name of a if ||a'(u)|| = 1
3: What is the name of the special surface given by a(v)=(f(v),0,g(v)), f(v) not 0, rotated about z axis to form S={(f(v)cosu,f(v)sinu,g(v))|u in R, v in I}
2: The name of the curve whos evolute is the initial curve
2: Curvature k(s)=? (in terms of n and t) (use . for dot product)
2: If a is a plane param. by arc length, k(s) does not equal 0, then the radius of curvature is given by?
3: Moving frame equations also known as?
QuestionAnswerExtra Info.
2: Which vector do we obtain by rotating t(s) by pi/2
3: f:U->R smooth and gradf(p) does not equal 0, then p is called what?
2: Curvature of a curve a=(x(s),y(s)) (param. by arc length)
2: The name of the curve traced by the centres of curvature
2: Where is the evolute singluar?
3: The name of the plane through a(s) spanned by t(s) and n(s)
1: If a'(u) does not equal 0 for all u in I then a is this type of curve
3: The name of the plane through a(s) spanned by n(s) and b(s)
3: Binomial vector defined by this formula (give both sides of equation)
2: Name of the point of a curve a where k(u)=0
3: if h(s0)=0, h'(s0)=0,..., h^(k-1)(s0)=0 then we say a has at least ? with the plane
3: What is given by this formula: ||a'xa''||/||a'||^3?
3: If c is a regular value, then f^-1(c) is a?

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