Solutions in differential forms

[shankarvn]

Is there any book in exterior algebra and differential forms which has problems worked out..ie solutions manual which comes along with the book?

 

[carbis]

I don't know, but here I want to add myself as interested (in such a book)!

 

[mathwonk]

i do not know of any. at that level of sophistication, people are usually less interested in someone else's homework solutions, and just do their own.

the only possible idea i have is a schaum's book. they always work problems out for you.

 

[rdt2]

There's a small section at the end of the Vector Analysis book in REA's 'Problem Solver' series that has worked examples. Bressoud's 'Second Year Calculus' (Springer) might also be useful.

However, I haven't seen anything that has a lot of the type of example that I think you and I are both looking for.

 

[mathwonk]

i myself became happy with differential forms by reading one of the elementary articles by harley flanders in a book on global differential geometry edited by s.s. chern, in the ams series. instead of talking abstractly about them he just showed how easy they were to calculate with.

actually they are so easy to calculate with, there is almost nothing to it.

let v,w,u,z be a vector basis for a 4 dimensional space say. then an alternating 2 vector on that space looks like a v^w + bv^u + c v^z + d w^u + e w^z + f z^u.

to multiply it by a 1 vector such as v, we just get
a v^v^w + b v^v^u + c v^v^z + d v^w^u + e v^w^z + f v^z^u. Now we cancel all those with 2v's in them, (or two of anything). and we get
d v^w^u + e v^w^z + f v^z^u.

similarly, to multiply that same gadget by the 2-vector v^w, just gives us

fv^w^z^u.

that's basically why there is only a one dimensional space of alternating 4 - vectors on a 4 dimensional vector space. also v^w = -w^v.

then one can do this same construction on the dual spaces. i.e. if x, y, z are the linear coordinates on three space, then they are also the absis of the dual space.

so expressions like 14 x^y^z, give alternating 3- co-vectors.

a 1 form is an assignment of an alternating 1- covector to each point. the standard ones are called dx, dy, dz, which assign to each point the covectors x, y, and z respectively.\

a general 1 form on euclidean 3 space looks like pdx + qdy + rdz where p,q,r, are functions. they are multiplied in the same way as above.

for example the "curl" of the 1 form pdx + qdy +rdz is defined as

dp^dx + dq^dy + dr^dz =

(?p/?x dx + ?p/?y dy + ?p/?z dz)^dx (those ?'s were curly d's when i entered them)

+ (?q/?x dx + ?q/?y dy + ?q/?z dz)^dy

+ (?r/?x dx + ?r/?y dy + ?r/?z dz)^dz

= now kill all the dx^dx and dy^dy and dz^dz terms, and change all the dy^dx terms to dx^dy, and so on, and you get

(?q/?x - ?p/?y)dx^dy + (?r/?y - ?q/?z)dy^dz + (?p/?z-?r/?x)dz^dx.

if you take another derivative of this (now called the "divergence") you should get zero, because of equality of mixed partials. see if you do.

i.e. the divergence of the 2 form Pdx^dy + Qdy^dz + Rdz^dx

= dP^dx^dy + dQ^dy^dz + dR^dz^dx, expanded as above.

this stuff is tedious but trivial, i.e. entirely mechanical.

here is another exercise: if u,v,w, are vectors and the coefficients are scalars, then (au+bv+cw)^(du+ev+fw)^(gu+hv+iw) = det(A)(u^v^w), where A is the matrix whose columns, or rows, are the cefficients vectors (a,b,c), (d,e,f), (g,h,i).

so all these guys are is a device for rendering determinant calculations mechanical. which is why they are both so tedious and useful.

here is a used copy of flanders article, or rather of chern's entire ams volume:

Chern, S.S., ed
Global Differential Geometry: Studies in Mathematics Vol. 27
Math. Assn of Am, 1989 1st, Hard Cover, fine/-- ISBN:0883851296
ISBN: 0883851296
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