# Exterior product

1. Jan 8, 2005

### vincentchan

What is exterior product?

2. Jan 8, 2005

### dextercioby

The short version is,where else,than here

NOTE:The ordinary cross product between vectors in E_{3} is a particular case of exterior/wedge product.

Daniel.

3. Jan 8, 2005

### vincentchan

I know its definition, my question is why is it define this way, ie. v^w=-w^v and v^v = 0

4. Jan 8, 2005

### dextercioby

Nope,that is his definition.It's like asking why is the derivative defined by
$$f'(x)=:\lim_{\Delta x\rightarrow 0}\frac{f(x+\Delta x)-f(x)}{\Delta x}$$
...That's the definiton and nothing more.Its properties follow as logical consequences from its definition.You can define other form of products,but the definition of exterior product is the same,no matter what happens...

Daniel.

5. Jan 8, 2005

### vincentchan

May I moldify my question the other way, What is exterior product use for and when will we use it (for a real physical application)?

6. Jan 8, 2005

### dextercioby

AAAAAAAAAAAhhhhhh...Yous should have said that from the beginning.Depends on the physics courses u take and the course material the student is being presented with.Any application into phyiscs of differential geometry will use "manifolds,cotangent spaces,p-forms and wedge products".General relativity,classical mechanics,quantum mechanics,thermodynamics.However,i say again,it's possible that these mathematical notions not be encountered:
For exampleamiltonian dynamics can be taught without the studying the symplectic manifold called "phase space".
General Relativity can be taught without geometrical description.Take for example,Dirac's course on GR.No manifolds,no p-forms,no Hodge duality,no nothing...
Geometric quantization can be taught only if you're taught classical dynamics and symplectic manifolds,like "phase-space".
Thermodynamics can be taught without making too much mathematical analysis of the differential forms like differential heat and differential work.

Again,it depends on the teacher.If he keeps a mathematically rigurous course,you'll see them.If not,not.

Daniel.

7. Jan 8, 2005

### marlon

Interesting, yet hard to believe...how is this done...please enlighten us...

mmm, it's a little bit more then that...Besides it is even better if you do not have any notion on the way classical dynamics work because it must be easier to not get confused...But your list is faaaaaaaaaaar from complete that it is wrong. Also incluse GTR + QFT Thus also QM and special relativity...Ever saw a paper on LQG, check out my journal and you shall be convinced ???

Then how ??? No partition functions, no total differentials (how are you gonna incorporate reversal processes and therefore hysteresis???) ??? What part of thermodynamics are you planning on teaching ???

CONCLUSION : i am glad not to be a student of yours

marlon

8. Jan 8, 2005

### dextercioby

I wrote there.I was referring to the course of Mr.Dirac.Why didn't u read all the way??Complete reading sometimes "enlightens"... :tongue2:

Who said i was making a complete list??Marlon don't read too much into it...You like twisting the words till they come your way...You have developped an awkward and unhealthy habit of imposing your point... :yuck:
Doesn't work with me,though...

Ever heard of the two formulations of classical reversible processes thermodynamics??Guess not,if u did,u know i was referrng to the neogibbsian formulation...Too bad u'll never be one of my students...You could learn so many...

Have it your way...I hope it gives you satisfaction... :tongue2:

Daniel.

9. Jan 8, 2005

### marlon

No no, that is avoiding an answer...Please explain how this is done...What is the content of this "course" that you obviously followed through completely...I am not asking for your opinion; i am asking for facts: explain how GTR can be derived whithout any geometrical models...

I certainly didn't...

Twisting words is NOT the same as correcting your mistakes, dear friend. Don't be so easily offended...

Sure, and how do you think the first law of thermodynamics is formulated ? What mathematical concepts are used ??? Maybe total differentials ??? i don't know ??? And what do these total differentials mean and represent physically ???

What said anything about Gibbs here ??? Please do not bring in topics that have NOTHING to do with this...Let's stick to the facts at hand

:rofl: :rofl: :rofl: :rofl: :rofl:
Nothing is more practical then a good theory...ofcourse you first need to understand it...
marlon

10. Jan 8, 2005

### hypermorphism

With respect to differential geometry, it's defined that way in order to have an algebraic expression for oriented volume. You can easily work backwards from the determinant to find the simplest expression for oriented volume from vectors as the definition of the exterior product.

11. Jan 8, 2005

### dextercioby

Section "5" from Mr.Dirac course is called "CURVED SPACE" and is:

Marlon,if that's a way to describe curved space-time in GR,then i'm i gonna go to bed... :yuck: I'm not saying it's ballooney,it's that it is outrageously simplistic and mathematically unfounded.

Marlon,the differential form of the first principle in CTPCN formulation contains indeed 2 one-forms and one exact differential.The problem was that i assumed (correctly,believe me,personal experience) that students were told:"Hey,this is not a exact total differential."And followed a rather simplistic explanation not once mentioning the words:"manifold,cotangent space,one-form".
I specifically said:
By "mathematical analysis" i obviously meant "differential geometry".Telling the students:"$dU$is a total differential,but $\delta Q$ and $\delta L$ aren't "is not diferential geometry,by no means...

I see Gibbs' work provokes you the same feeling of "uncertainty" as do the words "experimental physics" to me...
For your knowledge,Gibbs rebuilt thermodynamics and took over from where Boltzmann had placed statistical physics.

I see you're pretty tired and took 'than' and 'then' as the same thing... :tongue2:

Daniel.

12. Jan 8, 2005

### mathwonk

back to the original question: a wedge product is merely a coordinjate free way to represent a determinant, or a combination of determinants, hence is a way of measuring lengths, or areas, or volumes, or higher such,...., possibly by integrating over regions of space, possibly curved.

13. Jan 9, 2005

### marlon

Ohh come on, and this is no differential geometry ???

This is nonsense and pure speculation...

Oh yes it really is...

Sorry bit this a wrong conclusion

But really, where do you get these false arguments. What you say here is just plain wrong man !!!

marlon

ps : i suggest we drop this discussion because it has degenerated into speculations on maths and thermodunamics...