Hello,
It is well known any curve in projective plane, projective space etc. can be exactly defined by 1 parameter linear differential equations. Alternatively, they can be defined uniquely by giving equation between their torsion and curvature , or between relative differential invariants.
Surfaces in projective space can be defined by 2 parameter 2 linear differential equation system . For example, quadrics (second order surfaces) are defined by canonical system:
Xuu+bXv+fX=0
Xvv+aXu+gX=0
Where X(u,v) is quadric, u,v- parameters.
Taken separately, each of them represents a 1D Shrodinger wave equation in a "potential field " where one parameter plays role of time:
bXv+fX=-Xuu
v- time here. Solution of Shrodinger time dependent 1 D equation is thus a special case of quadric equation ( defined by coefficient relations) in a complex projective space .
Of course, these equations have to be looked at together with paired one of quadric to get full picture.
I am looking for a reference where the possible differential equation definitions of Klein Bottle in projective 3D space are exactly written out, preferably in canonical form (so in a coordinate system that leaves only the absolutely nesessary terms of the diff equation).
My aim is to understand to what physical process these differential equations of Klein bottle correspond if one parameter is replaced with time.
You are abusing nearly every polysylabic math word in the OP.
A Klein bottle is a topological, 2-D object. To plot it, you need a higher dimensional space in which to embed the surface. R^3 won't do it. And, as it turns out, RP^3 won't do it. http://www.univalle.edu.co/~osperdom/klein.pdf
One possible algebraic description of a Klein bottle embedded in R^4 is
w^2 + x^2 + y^2 + z^2 = 1
2wyz - x(y^2 + z^2) = 0
(Any mistakes in the above are mine. I haven't checked to see if the unique solution is a single klein bottle.)
Then you solve the equations respecting that you will have two degrees of freedom remaining. Every surface admits multiple parameterizations, and one that solves the above equations is:
w = cos 2u cos v
x = sin 2u cos v
y = cos u sin v
z = sin u sin v
(Thanks to H.B. Lawson, who in 1970 discovered and/or popularized this particular parameterization and embedding.)
But none of that is a differential description. Care to try again at honest, meaningful work?
A Klein bottle is a topological, 2-D object. To plot it, you need a higher dimensional space in which to embed the surface. R^3 won't do it. And, as it turns out, RP^3 won't do it. http://www.univalle.edu.co/~osperdom/klein.pdf
One possible algebraic description of a Klein bottle embedded in R^4 is
w^2 + x^2 + y^2 + z^2 = 1
2wyz - x(y^2 + z^2) = 0
(Any mistakes in the above are mine. I haven't checked to see if the unique solution is a single klein bottle.)
Then you solve the equations respecting that you will have two degrees of freedom remaining. Every surface admits multiple parameterizations, and one that solves the above equations is:
w = cos 2u cos v
x = sin 2u cos v
y = cos u sin v
z = sin u sin v
(Thanks to H.B. Lawson, who in 1970 discovered and/or popularized this particular parameterization and embedding.)
But none of that is a differential description. Care to try again at honest, meaningful work?
A Klein bottle can't be put into R^3 without self intersecting and thus it does not form a submanifold of R^3 and a differential approach is going to be close to, if not, impossible.
Besides Ivars, you don't know jack **** about vector calculus, never mind the differential geometry of non-orientated spaces. Do you really believe you can do such mathematics?
Besides Ivars, you don't know jack **** about vector calculus, never mind the differential geometry of non-orientated spaces. Do you really believe you can do such mathematics?
Thank You, both.
AN- no, I do not know vector calculus- neither did I know its needed to produce differential equation for a surface in projective space- Wylczinski , Halphen , Lane, Finnikov never mention a vector in their main works about projective differential geometry - they work in complex projective space CP^3, mostly with projective homogeneous coordinates, coordinate tetrahedrons, without referring to R^3 anywhere.
May be I have missed something, but to me it seems that vector concept is not needed there.
So You are saying that Klein bottle can not be expressed via differential equations whose solutions in projective space CP^3 is a surface we know as Klein bottle?
That answers my question.
AN- no, I do not know vector calculus- neither did I know its needed to produce differential equation for a surface in projective space- Wylczinski , Halphen , Lane, Finnikov never mention a vector in their main works about projective differential geometry - they work in complex projective space CP^3, mostly with projective homogeneous coordinates, coordinate tetrahedrons, without referring to R^3 anywhere.
May be I have missed something, but to me it seems that vector concept is not needed there.
So You are saying that Klein bottle can not be expressed via differential equations whose solutions in projective space CP^3 is a surface we know as Klein bottle?
That answers my question.
As I suspected, because tan a = tan 2b has the solution a = 2(b + pi/2), another constraint is needed:
2yz sqrt(1-y^2-z^2) = x(y^2 + z^2)
Or the solution contains a twinned Klein bottle.
Or you can abuse notation and write
w^2+x^2+y^2+z^1 = 1
(w,x) = +/- (y^2 - z^2, 2yz) sqrt(1-y^2-z^2)/(y^2 + z^2)
2yz sqrt(1-y^2-z^2) = x(y^2 + z^2)
Or the solution contains a twinned Klein bottle.
Or you can abuse notation and write
w^2+x^2+y^2+z^1 = 1
(w,x) = +/- (y^2 - z^2, 2yz) sqrt(1-y^2-z^2)/(y^2 + z^2)
QUOTE (rpenner+Sep 18 2009, 10:04 PM)
Or the solution contains a twinned Klein bottle.
Solution of this set of equations?
w^2 + x^2 + y^2 + z^2 = 1
2wyz - x(y^2 + z^2) = 0
So we have a 2 equation system with 4 variables in R4.
W1(w,x,y,z)=1
W2(w,x,y,z)=0
One day I will try to differentiate it as many times as necessary and see if that gives differential equations for twinned Klein bottle. Probably Klein had these equations somewhere in his original papers, or Lie.
Solution of this set of equations?
w^2 + x^2 + y^2 + z^2 = 1
2wyz - x(y^2 + z^2) = 0
So we have a 2 equation system with 4 variables in R4.
W1(w,x,y,z)=1
W2(w,x,y,z)=0
One day I will try to differentiate it as many times as necessary and see if that gives differential equations for twinned Klein bottle. Probably Klein had these equations somewhere in his original papers, or Lie.
QUOTE (Ivars+Sep 18 2009, 03:07 PM)
AN- no, I do not know vector calculus- neither did I know its needed to produce differential equation for a surface in projective space- Wylczinski , Halphen , Lane, Finnikov never mention a vector in their main works about projective differential geometry - they work in complex projective space CP^3, mostly with projective homogeneous coordinates, coordinate tetrahedrons, without referring to R^3 anywhere.
Do you think 'vector calculus' is only used for R^3?! That proves you've not got a bloody clue. Vector calculus is about considering dynamical systems in multiple variables and multiple dimensions. It can be in R^3, S^3, CP^5, AdS10, whatever. It's about describing the position and motion of tyhings. If you have coordinates then you have vector calculus.
Tell me, what books or lecture notes or journals have you been reading since you last posted in this thread? Or haven't you done any actual learning?
Do you think 'vector calculus' is only used for R^3?! That proves you've not got a bloody clue. Vector calculus is about considering dynamical systems in multiple variables and multiple dimensions. It can be in R^3, S^3, CP^5, AdS10, whatever. It's about describing the position and motion of tyhings. If you have coordinates then you have vector calculus.
Tell me, what books or lecture notes or journals have you been reading since you last posted in this thread? Or haven't you done any actual learning?
QUOTE (AlphaNumeric+Oct 20 2009, 01:21 PM)
Do you think 'vector calculus' is only used for R^3?! That proves you've not got a bloody clue. Vector calculus is about considering dynamical systems in multiple variables and multiple dimensions. It can be in R^3, S^3, CP^5, AdS10, whatever. It's about describing the position and motion of tyhings. If you have coordinates then you have vector calculus.
Tell me, what books or lecture notes or journals have you been reading since you last posted in this thread? Or haven't you done any actual learning?
Hi
I have read everything, like Sarah Palin, but no deeply enough, so I can not say I have read/studied, more like realized the existance.
Yes I now see the application of vectors to dynamical systems in many dimensions, especially related to speed at position in a flow plus the flow of vector field itself, together. Earlier I just did not see the need, sorry, and did no want to start with real coordinates ( I know that You can view CP1 as pair of real coordinates). I did not understand that any space itself can be made to flow and curve.
I still have some aversion to real vectors since they imply length, metric , but vectors in continuous complex projective spaces are OK for me, and I am slowly trying to understand their applications.
For example, if You have 2 speed vectors (e.g. tangential speed and axial linear speed of the body - (but as complex numbers) in some projective space at different points, and if the space itself allows finite translation of vectors , than summation of these 2 will give a resulting swirl vector properly in every point inside and on the vortex cone. That implies, that rotation+translation of this part of space happens as rigid body, I think, and that space itself is with rigid body properties. This could be applied to inner core of e.g. vortexes with swirl, forced vortex.
If the finite translation in the space however is not allowed, than infinitesimal might be, and that allows to use vector differential summation and intergration. So here infinitesimal parts of object move as rigid body, but the whole space might be curved.
I guess there are also spaces/processes which restrict the ways infinitesimal vectors can be transformed/added, and that leads to Lie transformation groups, but that topic is still not clear for me.
I feel its important to understand these infinitisemal vector transformations in projective spaces to understand possible symmetries of physical processes.
Tell me, what books or lecture notes or journals have you been reading since you last posted in this thread? Or haven't you done any actual learning?
Hi
I have read everything, like Sarah Palin, but no deeply enough, so I can not say I have read/studied, more like realized the existance.
Yes I now see the application of vectors to dynamical systems in many dimensions, especially related to speed at position in a flow plus the flow of vector field itself, together. Earlier I just did not see the need, sorry, and did no want to start with real coordinates ( I know that You can view CP1 as pair of real coordinates). I did not understand that any space itself can be made to flow and curve.
I still have some aversion to real vectors since they imply length, metric , but vectors in continuous complex projective spaces are OK for me, and I am slowly trying to understand their applications.
For example, if You have 2 speed vectors (e.g. tangential speed and axial linear speed of the body - (but as complex numbers) in some projective space at different points, and if the space itself allows finite translation of vectors , than summation of these 2 will give a resulting swirl vector properly in every point inside and on the vortex cone. That implies, that rotation+translation of this part of space happens as rigid body, I think, and that space itself is with rigid body properties. This could be applied to inner core of e.g. vortexes with swirl, forced vortex.
If the finite translation in the space however is not allowed, than infinitesimal might be, and that allows to use vector differential summation and intergration. So here infinitesimal parts of object move as rigid body, but the whole space might be curved.
I guess there are also spaces/processes which restrict the ways infinitesimal vectors can be transformed/added, and that leads to Lie transformation groups, but that topic is still not clear for me.
I feel its important to understand these infinitisemal vector transformations in projective spaces to understand possible symmetries of physical processes.
QUOTE (Ivars+Oct 22 2009, 07:15 AM)
I have read everything, like Sarah Palin
You do realise that's a byword for "I read nothing". You're hardly putting yourself in a good position if you're comparing your learning style to Dipshit Palin. At least tell me you know Africa isn't a country, right?
You do realise that's a byword for "I read nothing". You're hardly putting yourself in a good position if you're comparing your learning style to Dipshit Palin. At least tell me you know Africa isn't a country, right?
QUOTE (Ivars+Oct 22 2009, 07:15 AM)
I still have some aversion to real vectors since they imply length, metric , but vectors in continuous complex projective spaces are OK for me, and I am slowly trying to understand their applications.
Again, you show your ignorance. You can define real projective spaces in precisely the same manner as complex ones and so RP^n has no notion of distance, so real vectors aren't synonymous with distances and lengths and metrics only exist in spaces where you can define norms and metrics, which is not a universal property. And if you don't understand real coordinates then you don't understand complex ones, they are more complicated and more subtle. If you understand complex spaces you understand real ones because you can obtain a real space from a complex one always because a complex space is a real one with a particular structure. But you should know this, if you weren't deluding yourself.
QUOTE (Ivars+Oct 22 2009, 07:15 AM)
For example, if You have 2 speed vectors (e.g. tangential speed and axial linear speed of the body - (but as complex numbers) in some projective space at different points, and if the space itself allows finite translation of vectors , than summation of these 2 will give a resulting swirl vector properly in every point inside and on the vortex cone. That implies, that rotation+translation of this part of space happens as rigid body, I think, and that space itself is with rigid body properties. This could be applied to inner core of e.g. vortexes with swirl, forced vortex. .
Wow, clearly your grasp of such things as curl and grad is amazing. Perhaps if you could actually do the algebra you'd be able to explain it in a little more of a coherent fashion. When I talk about differential geometry to people I use more algebra than I do words. You use no algebra. Months and months ago Euler challenged you to put up or shut up, betting that you wouldn't learn anything over 3 or 6 months and here we are, even further down the line and you've acheived nothing it would seem.
QUOTE (Ivars+Oct 22 2009, 07:15 AM)
I feel its important to understand these infinitisemal vector transformations in projective spaces to understand possible symmetries of physical processes.
I feel it's important to actually be able to do the maths to understand it. But what would I know about that?
QUOTE (AlphaNumeric+Oct 23 2009, 04:25 PM)
Again, you show your ignorance. You can define real projective spaces in precisely the same manner as complex ones and so RP^n has no notion of distance, so real vectors aren't synonymous with distances and lengths and metrics only exist in spaces where you can define norms and metrics, which is not a universal property.
When You have a complex curve that has real and imaginary parts, to use Real vector space means to use 6D space instead of 3D.
While 3D is easy to visualize, 6D a bit hard.
Even for Complex plane curve, You need 4D real vector space.
The only thing one can visualize from complex geometry in this way, is a complex point, placed on real plane, Argand plane.
In Projective complex space, even curve in CP3 can be visualized by using 2 types of lines for its real and imaginary parts.
In this way, also 6 dimensional Real curves can be visualized, for example, sextics , in CP3. Quintics and quartics as well.
I really do not understand the benefits of creating non-visualizable spaces out of relatively simple CP3 or CP2.
When You have a complex curve that has real and imaginary parts, to use Real vector space means to use 6D space instead of 3D.
While 3D is easy to visualize, 6D a bit hard.
Even for Complex plane curve, You need 4D real vector space.
The only thing one can visualize from complex geometry in this way, is a complex point, placed on real plane, Argand plane.
In Projective complex space, even curve in CP3 can be visualized by using 2 types of lines for its real and imaginary parts.
In this way, also 6 dimensional Real curves can be visualized, for example, sextics , in CP3. Quintics and quartics as well.
I really do not understand the benefits of creating non-visualizable spaces out of relatively simple CP3 or CP2.
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