Problem: Show that the set S={u=(x,y,z) in R^3 | 2x^2 + y^2 + z^2 = 1} is pathwise-connected.

I plotted the function in MATLAB and attached it. In a way, this looks very similar to the unit sphere in $\displaystyle R^3$, which is pathwise connected where the parametric path is

f(t) = costu+sintw

where w is a point in the sphere that is perpendicular to u and v = -u. However, I am having a hard time to find a parametric path for this set.

The book says to use two previous exercises 1.) The union of two pathwise-connected subsets of R^n is pathwise-connected and 2.)the graph of the mapping F:A -> R^m is continuous then if A is pathwise-connected, then the set

G = {(u,v) in R^(n+m)|u in A, v = F(u))} is pathwise-connected.

Image:

Any help is greatly appreciated. Thank you.