let (x,d) be a metric space. define a flow on (x,d)

If a vector function F depends on both space coordinates (x, y, z) and time t, show that:

An _____ is a set of points (x, y) in a plane such that the sum of the distances between (x, y) and two fixed points called _____ is a constant.

the usual metric space defined by  d(x,y)= x-y prove the four propertis of metric space

Suppose that (X,d) is a metric space and R has the standard metric. Let f : X → R be a continuous function. For each r > 0, show that A={x∈X| |f(x)|<r} is open in X.

State true or false with a brief justification If the dual X' of a normed linear space X is fininte dimensional, then X is finite dimensional

Metrics. Show that, for fixed x0 in E, the function  from E into ℝ+ is uniformly continuous.  

1. a) Let (x, d) be a metric space. Define a flow on (x, d). b) Let (x, {ϕt}) be a flow on a metric space X. When is xo in x a fixed point of the flow? c) When do you say that a fixed point xo in x is Poincare stable? d) When do you say that a fixed point xo is Lypanov stable?

calculate div(F) and curl(F).  F = (x, y, z}

Prove that topological space E is not homeomorphic to the spaceY = {(x, y) ∈ E^2 : y = ± x} (E represents R equipped with Euclidean distance, E^2 represents R^2 equipped with euclidean distance)