differential problem

May 2009
45
1
a catenoid can written parametrically as

p=(c*cosh(u/c)*cos(v), c(cosh(u/c)*sin(v),u)

a. find dx/du|p and dx/dv|p

im confused is x the x component of p? or is p meant to be x(p) in which differentiate all of it... i think i am as all my other example have x(p), could anyone just say if thats what im supposed to do?
 

Bruno J.

MHF Hall of Honor
Jun 2009
1,266
498
Canada
You have \(\displaystyle p(u,v)=(x(u,v),y(u,v),z(u,v))\); you are probably asked to find \(\displaystyle \frac{\partial x}{\partial u} \Big |_{(u_0,v_0)}\). It doesn't make much sense to write \(\displaystyle \frac{\partial x}{\partial u} \Big |_{p}\).
 
May 2009
45
1
ok then so \(\displaystyle
\frac{\partial x}{\partial u} \Big |_{p}
\)=(sinh(u/c)cos(v),sinh(u/c)sin(v),1)

and \(\displaystyle
\frac{\partial x}{\partial v} \Big |_{p}
\)=(-c*cosh(u/c)sin(v),c*cosh(u/c)cos(v),0)

then E=c^2*cosh^2(u/c)
F=0
G=cosh^2(u/c)

is that right?
 

Bruno J.

MHF Hall of Honor
Jun 2009
1,266
498
Canada
No! Partial derivatives are always real-valued, not vector-valued. You have to differentiate \(\displaystyle x(u,v)\) with respect to \(\displaystyle u\), not form the vector \(\displaystyle (\partial x / \partial u, \partial y / \partial u, \partial z / \partial u)\).
 
May 2009
45
1
ok this is where I'm getting confused... because the partial derivatives on my sheet are vectors? as is the x, in dx, has a tilda underneath it??? doesnt this mean that p should be x(p), where they both are vectors?
 

Bruno J.

MHF Hall of Honor
Jun 2009
1,266
498
Canada
ok this is where I'm getting confused... because the partial derivatives on my sheet are vectors? as is the x, in dx, has a tilda underneath it??? doesnt this mean that p should be x(p), where they both are vectors?
I do not understand what you are talking about. As far as I'm concerned, the expression \(\displaystyle x(p)\) has no meaning whatsoever in this context. Did you copy the problem exactly? Are you sure \(\displaystyle p\) denotes a point of the surface and not a point of the \(\displaystyle u,v\) domain which parametrizes it? This would explain the notation \(\displaystyle \partial x/\partial u |_p\).

In any case, I can tell you that partial derivatives are not vectors!
 
May 2009
45
1
let me say the question in full.

A catenoid is a surface of revolution, points p(tilda) belongs to C, can be parametrized as p(tilda)=(c*cosh(u/c)cos(v),c*cosh(u/c)sin(v),u)

calculate dx(tilda)/du|p and dx(tilda)/dv|p, the basis vectors for the tangent space to C at p(tilda),Tp(C)

does that make sense to you? sorry but my lecturer taught me lin alg last yr, and since diff geo lecturer died, he took over... my lecturer is pretty hopeless, and doesnt know much about this topic.. so its really hard to learn from him.
 

Bruno J.

MHF Hall of Honor
Jun 2009
1,266
498
Canada
let me say the question in full.

A catenoid is a surface of revolution, points p(tilda) belongs to C, can be parametrized as p(tilda)=(c*cosh(u/c)cos(v),c*cosh(u/c)sin(v),u)

calculate dx(tilda)/du|p and dx(tilda)/dv|p, the basis vectors for the tangent space to C at p(tilda),Tp(C)

does that make sense to you? sorry but my lecturer taught me lin alg last yr, and since diff geo lecturer died, he took over... my lecturer is pretty hopeless, and doesnt know much about this topic.. so its really hard to learn from him.
It would definitely have helped to know that this is a differential geometry problem and not a multivariable calculus problem, as the expressions involved have quite different meanings in both contexts. Please quote the whole problem in the first post next time; it'll save both of us time.

Right now I'm going to bed but I'll get back to you tomorrow if I can.