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Differential Geometry | Topology

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 x_{0}. Prove "\\kappa" (s_{0}) > 1. (Hint: Consider f(s) = ||x(s)|| ^{2} . Then f(s) has a local maximum at s_{0}. Calculate f''(s_{0}))

Differential Geometry | Topology

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).

Differential Geometry | Topology

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

Differential Geometry | Topology

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

Differential Geometry | Topology

Define polar developable

Differential Geometry | Topology

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

Differential Geometry | Topology

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.

Differential Geometry | Topology

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

Differential Geometry | Topology

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

Differential Geometry | Topology

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