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# Differential Geometry | Topology Answers

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Let x(s) be a curve with arc length parametrization, and satisfies "||x(s)||\\leq ||x(s_0)||\\leq1" for all s sufficiently close to x0. Prove "\\kappa" (s0) > 1. (Hint: Consider f(s) = ||x(s)|| 2 . Then f(s) has a local maximum at s0. Calculate f''(s0))

Let f be a smooth function. Calculate the curvature and the torsion of the curve that is the intersection of x = y and z = f(x).

Calculate T ; N; B; "\\kappa" ; "\\tau" of the curve x(t) = (t; t 2 ; t 4) at the point (1; 1; 1).

prove that a neccessary and sufficient condition Bor a surface to be dovelopable us that its Gaussian curvature is zero

Define polar developable

Given that vector A=3i + j + k, B=2i - j +2k and C=i + j + k

(a) Find a unit vector normal to the plane containing vector A+B and A+C

(b) Find the unit vector normal to the plane containing vector A+(A+B)B and C

(c) Why is the unit vector normal to the plane containing A and B parallel to the vector normal plane containing (A•B)A and (B•C)B

let X = {a, b, c} and tau = {pi, X, {a}, {b, c}} and let A = {a} be the subset of the topological space (X, tau). Find the interior and closure of A. explain in detail.

3. Prove that A space curve is a helix if and only if the ratio of the curvature to the torsion is

constant at all points

Prove that a space curve is a helix if and only if. The ratio of the curvature to the torsion is constant at all points

Find the instrinsic equation of the curve by x=ae^ucosu; y=ae^usinu; z=be^u

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