Ok... some hints.
a) Show for all . A basis for is .
To show that , express as a linear combination of the basis vectors and show this cannot be.
b) Compute and show it is nonzero for all .
c) Check that , so that the tangent space contains .
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.
Ok... some hints.
a) Show for all . A basis for is .
To show that , express as a linear combination of the basis vectors and show this cannot be.
b) Compute and show it is nonzero for all .
c) Check that , so that the tangent space contains .