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?

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