1. ## reparametrization HELP!!!

PLEASE CAN ANYONE HELP ME?? I am completely stuck on this geometry!! If there is a geometry-genious out there: THANK YOU!!
**i have attached the original question sheet!**

(sorry, had no maths symbols, so note: E stands for "element of". And f is supposed to be SIGMA, and g is supposed to be TAU)

Let U = {(u, v) E R2 | 0 < v < u < 1} and
f(u, v) = (u, u − v, u^2 + v)
for (u, v) E U.

a. Show that f is a regular plane, and determine a basis for the tangent space Tpf, for every p E U.
Prove, that there does not exist a p E U for which e1 E Tpf . Where e1 is the standard basis vector (1,0,0) in R3.

b. Let
W =]0, 1[×]0, 1[= {(s, t) E R2 | 0 < s < 1, 0 < t < 1} and

g(s, t) = (s, s − st, s^2 + st)

for (s, t) E W. Show that g is a reparametrization of f.

c. Let

p(s, t) = (s2 + t, t, st^2)
for (s, t) EV , where V is an open interval, which includes the pointq = (1/2, 0 ). Show that e1 ETqp and show how p cannot be a reparametrization of f.

2. Ok... some hints.

a) Show $|| f_u\times f_v||\neq 0$ for all $(u,v)$. A basis for $T_pf$ is $\{f_u(p),f_v(p)\}$.

To show that $e_1\notin T_pf \ \forall p$, express $e_1$ as a linear combination of the basis vectors and show this cannot be.

b) Compute $\frac{\partial(u,v)}{\partial(s,t)}(s,t)$ and show it is nonzero for all $(s,t)$.

c) Check that $(P_s\times P_t)(q)=(0,0,1)$, so that the tangent space $T_qP$ contains $e_1$.