Differential Geometry | Topology Answers

Sketch the level curves f−1(c) for the following functions: (a) f(x,y) = x2 + y2, c = 0,1,2,3,4.

If u=3t2i+et j−t k and v=sinti−costj+e2tk then, at t=0, ddt(u×v)=

Find the unit tangent T, the unit normal N and the curvature K for the curve r(t) =<2 sint, 5t, 2 cost>.

Find the curvature of the parabolic curve r(t) =<t,t^2>.

Reparameterize the curve r(t) = <2t,1-3t, 5 + 4t> with respect to the arclength s, measured from the point t= 0.

Show that if X is Hausdorff, a net in X converges to atmost one point.

Prove that an infinite product of discrete spaces may not be discrete.

Prove
1. If X is connected, then every quotient space of X is connect .
2. If X is compact ,then every quotient of X is compact.

Prove that a space X is homeomorphic to an open subspace of a compact Hausdorff space if and only if X is locally compact.

Show that Hilbert space is not locally compact at any point.