91Ó°ÊÓ

Let \(S\) be a regular surface covered by coordinate neighborhoods \(V_{1}\) and \(V_{2}\). Assume that \(V_{1} \cap V_{2}\) has two connected components, \(W_{1}, W_{2}\), and that the Jacobian of the change of coordinates is positive in \(W_{1}\) and negative in \(W_{2}\). Prove that \(S\) is nonorientable.

Show that the cylinder \(\left\\{(x, y, z) \in R^{3} ; x^{2}+y^{2}=1\right\\}\) is a regular surface, and find parametrizations whose coordinate neighborhoods cover it.

Let \(S_{2}\) be an orientable regular surface and \(\varphi: S_{1} \rightarrow S_{2}\) be a differentiable map which is a local diffeomorphism at every \(p \in S_{1}\). Prove that \(S_{1}\) is orientable.

Show that the equation of the tangent plane of a surface which is the graph of a differentiable function \(z=f(x, y)\), at the point \(p_{0}=\left(x_{0}, y_{0}\right)\), is given by $$ z=f\left(x_{0}, y_{0}\right)+f_{x}\left(x_{0}, y_{0}\right)\left(x-x_{0}\right)+f_{y}\left(x_{0}, y_{0}\right)\left(y-y_{0}\right) $$ Recall the definition of the differential \(d f\) of a function \(f: R^{2} \rightarrow R\) and show that the tangent plane is the graph of the differential \(d f_{p}\).

Show that the paraboloid \(z=x^{2}+y^{2}\) is diffeomorphic to a plane.

Show that the two-sheeted cone, with its vertex at the origin, that is, the set \(\left\\{(x, y, z) \in R^{3} ; x^{2}+y^{2}-z^{2}=0\right\\}\), is not a regular surface.

Construct a diffeomorphism between the ellipsoid $$ \frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}+\frac{z^{2}}{c^{2}}=1 $$ and the sphere \(x^{2}+y^{2}+z^{2}=1\).

Show that the area \(A\) of a bounded region \(R\) of the surface \(z=f(x, y)\) is $$ A=\iint_{Q} \sqrt{1+f_{x}^{2}+f_{y}^{2}} d x d y $$ where \(Q\) is the normal projection of \(R\) onto the \(x y\) plane.

Let \(f(x, y, z)=(x+y+z-1)^{2}\). a. Locate the critical points and critical values of \(f\). b. For what values of \(c\) is the set \(f(x, y, z)=c\) a regular surface? c. Answer the questions of parts a and b for the function \(f(x, y, z)=\) \(x y z^{2}\).

The coordinate curves of a parametrization \(\mathbf{x}(u, v)\) constitute a Tchebyshef net if the lengths of the opposite sides of any quadrilateral formed by them are equal. Show that a necessary and sufficient condition for this is $$ \frac{\partial E}{\partial v}=\frac{\partial G}{\partial u}=0 $$