A catenoid is a surface of revolution. Points in a cateniod, p e C, can be written parametrically as
x~(p~)=(c*cosh(u/c)cos(v),c*cosh(u/c)sin(v),u)
a. calculate dx~/du|p and dx~/dv|p
dx~/du|p=(sinh(u/c)cos(v),sinh(u/c)sin(v),1)
dx~/dv|p=(-c*cosh(u/c)sin(v),c*cosh(u/v)cos(v),0)
b. If a metric on C is defined by g(X,Y):=^(3)g(X,Y), where X,Y are vector fields tangent to C, show that the components g'ij of g relative to the basis {dx~/du,dx~/dv} for vector fields on C are
[g'ij]=[cosh^2(u/c) 0
0 c^2*cosh^2(u/c)]
E=||dx~/du||^2=(sinh^2(u/c)cos^2(v))+(sinh^2(u/c)sin^2(v))+1^2=cosh^2(u/c)
F=dx~/du.dx~/dv=0??? ive tried to do the math, but cant get zero, but all
G=||dx~/dv||^2=(-c^2*cosh^2(u/c)sin^2(v))+(c^2*cosh^2(u/c)cos(v))=c^2(cos^2(v)+sin^2(v))cosh^2(u/c)=c^2*cosh^2(u/c)
e=1/c
f=0
g=-c
I cant find out how to find these values properly, coz according to my maths wolfram and other are wrong, they a back to front from what i calculated?
c.Use the metric of part b. in Gauss's formula to calculate K(p) e H.
K(p)=eg-f/(EG-F)=-1/(c^2*cosh^4(u/c))=-sech^4(u/c)/c^2
d. Using the metric of part b. show that the only non-zero
are
=
=
=tanh(u/c)/c and
=-c*tanh(u/c). Hence write the geodesic equations on the catenoid. Dont solve them
=1/2g^11{dg11/du+dg11/gu-dg11/du}=1/2{2*cosh(u/c)sinh(u/c)/c}=tanh(u/c)/c
then form there i have no idea... and i think the other questions are quite wrong...
e.are the v-coordinate curves geodesic curves? [hint use your geodesic equations from d. above]