Differential Forms: What You'll Learn in Differential Geometry

[quasar987]

What are differential forms?

Is this what I'm going to learn about in my upcoming differential geometry class?

 

[dextercioby]

Yes,it's one part of this HUGE domain that we call differential geometry.They become really important once you consider physical phenomena,though mathematicians like to think they're important in mathematics as well.

So,either take the entry on wikipedia on "diff.forms" or wait till your teacher presents them in class...

Daniel.

 

[bombadillo]

A classical course on differential geometry -- which includes many introductory courses -- may not cover and use differential forms at all.

 

[quetzalcoatl9]

A classical course on differential geometry -- which includes many introductory courses -- may not cover and use differential forms at all.

This is true. Infact, many folks who do traditional vector calculus are unaware of differential forms (which is too bad because they are the ones who would probably benefit the most from them).

The short (and least satisfying) answer is that an n-form is an antisymmetric tensor of rank (0,n) that enjoys some nice symmetry properties - these properties essentially make the indices of the tensor "disappear". They can be thought of as functions that take tangent vectors as input They are also well defined across something called "pullback", something that is not true of tensors in general, so in effect they transcendent tensors. They also can provide topological information.

Try doing a search on this forum and you will see previous discussions. If you are math oriented, then there are several good modern differential geometry texts that cover the topic. If you are an engineer or in some applied science, and just wish to be able to use them without worrying about all of the details, then I can recommend the book by Weintraub (sp? it may be Weintraube, I forget, he is a professor in Louisianna) called "Differential Forms: A Complement to Vector Calculus" or something like that.

 

[bombadillo]

The short (and least satisfying) answer is that an n-form is an antisymmetric tensor of rank (0,n) that enjoys some nice symmetry properties - these properties essentially make the indices of the tensor "disappear". They can be thought of as functions that take tangent vectors as input They are also well defined across something called "pullback", something that is not true of tensors in general, so in effect they transcendent tensors. They also can provide topological information.

Tangent vectors "push forward", whereas forms "pull back", in differentiable mappings from one manifold to another. The topological information comes from the de Rham cohomology ring.

... then I can recommend the book by Weintraub (sp? it may be Weintraube, I forget, he is a professor in Louisianna) called "Differential Forms: A Complement to Vector Calculus" or something like that.

Weintraub, published by Academic Press. Traditional vector calculus finds a more general and elegant expression in the language of forms, and at some point one should see Stokes' theorem in this language. If you introduce a dot product on a manifold, then these general results reduce to the usual results of classical vector analysis.

 

[dextercioby]

Weintraub's book is really nice. I would have considered Spivak's book "Calculus on Manifolds",too,especially since it starts at a lower level,proving many results of "ordinary calculus".

Daniel.

 

[mathwonk]

when is someone going to reveal that there is an entire thread devoted to this topic, called "a geometric approach to differential forms" by david bachman, just below here, with zillions of entries and a free book?

 

[quetzalcoatl9]

when is someone going to reveal that there is an entire thread devoted to this topic, called "a geometric approach to differential forms" by david bachman, just below here, with zillions of entries and a free book?

Try doing a search on this forum and you will see previous discussions.<snip>

it has been mentioned. i saw no reason to limit the discussion to that single thread when there are tons of threads on the forum on this subject.

how many times will someone pop on and ask (without even doing a simple google search or something) "what is a tensor?" or "what is a differential form?", never to be heard from again, while everyone else starts chiming in interesting points, leading to an enlightening discussion? :)

 

[mathwonk]

here is an actual answer:

given a positive integer n, consider the generators dx1,...dxn, and define a product on them by (dxj)^2 = 0 and (dxj)(dxi) = -(dxi)(dxj).

then call the space of all R linear combinations of products the symbols dx1,...,dxn, the linear space of alternating forms (at a point). These form a graded algebra under multiplication graded by the number of the basic symbols appearing.


e.g. dx1dx2 + dx1dx3 is alternating 2-form.

If C is the space of smooth functions on R^n, call the space of all C linear combinations of these same symbols, and of thje constant function 1, the space of differential forms on R^n. This too is graded and the piece of degree r is the space of differential r forms.

e.g. C is the space of differential zero forms

there is a differentiation on this algebra defined as follows:

if f is a smooth function then df is the one form df = ?f/?x1 dx1 +...?f/?xn dxn,

[these curly partial derivatives will probably disappear on this site]

the derivative of f dx1dx2...dxj is the product dfdx1dx2...dxj.

extending this lienarly gives the (exterior) derivative of any form.

then on R^3 say, we have 1 basic 0 form, 1, and one basic 3form dxdydz, whilke we have three basic 1 forms and three basic 2 forms, dx,dy,dz, and dxdy, dydz, and dzdx.


with these definitions the derivative of a 1 form is what used to be called the curl, back in the nineteeth century, [actually "rot" for rotation, by maxwell], and still is by people who do not know about diff forms.

similarly the derivative of a 2 form was called the divergence, and the derivative of a function was called the gradient.

these archaic terms actually convey information not captured in the purely formal definition of exterior derivative, as one learns in the physics interpretation of the basic theorem: that the integral of a form over the boundary of a region equals the integral of its derivative over the interior of that region.

i.e. in physics one interprets these forms as dual to various flows of liquid in which case the idea of rotation and divergence become meaningful.