Help with vector identities

Joe09 · #1 by **Joe09** » Jan 08, 2012 6:19 pm

Hello there, as part of my advanced vector calc course i have to be able to prove certain vector identities with and without suffix notation

Personally ive never been any good at pure math and i am struggling with these questions and so have turned for help

this is going to be a long post as ill show my current workings aswell, x = cross product and . = dot product naturally

Question 1: show that grad x(A x B) = (grad . B)A - (grad . A)B + (B . grad)A - (A . grad)B where (A . grad)B is defined as the vector [(A . grad)Bx, (A . grad)By, (A . grad)Bz]

Working out so far:

realised mistake so working not in anyway useful, sigh guess i keep working on it, and pointers would be great

Question 2: show that grad.(A x B) = B.(grad x A) - A.(grad x B)

working out so far:

LHS

(A x B)x = AyBz - AzBy

so gradx.(A x B)x = d/dx(AyBz - AzBy)

RHS

(grad x A)x = (Az)d/dy - (Ay)d/dz
and similarly (grad x B)x = (Bz)d/dy - (By)d/dz

so Bx.(grad x A)x = Bx[(Az)d/dy - (Ay)d/dz]
and similarly Ax.(grad x B)x = Ax[(Bz)d/dy - (By)d/dz]

so Bx.(grad x A)x - Ax.(grad x B)x and rearranged =

d/dy(AzBx - AxBz) - d/dz(BxAy - ByAx)

and here is where i am stuck

Question 3: same as question 2 except that now i must prove it using suffix notation

working out so far:

(A x B)i = eijkAjBk

(grad x A)i = eijkAkd/dxj and (grad x B)i = eijkBkd/dxj

next B.(grad x A)i = BieijkAkd/dxj and A.(grad x B)i = AieijkBkd/dxj

so B.(grad x A)i - A.(grad x B)i = eijkd/dxj(AkBi - AiBk)

and im stuck from here

Question 4: use suffix and show that grad x grad x A = -(grad . grad)A + grad(grad . A) where (grad . grad)A is defined as [(grad . grad)A]i = grad2Ai = (Aid2/dx12 + ...)

working out so far

LHS

grad x (grad x A)i = eijkd/dxj(eklmAmd/dxl

= Am(&il&jm - &im&jl)d2/dxjdxl

RHS

taking grad2Ai = Aid2/dxj2

and grad.(grad . A)i = d/dxi(d/dxj)Aj = Ajd2/dxidxj

any help you can give id appreciate very much

Joe09 · #2 by **Joe09** » Jan 08, 2012 6:23 pm

already noticed mistake in q1, ive done div not grad for LHS

cavarka9 · #3 by **cavarka9** » Jan 16, 2012 2:01 pm

not to distract you But i didnt do those back few yrs because its tedious but not difficult,

it would be better for you to jump into problems directly, that way you will know the methods that work, also write down imp and useful formula on piece of white sheet by your side, that way you can just get on with your work without having to look back.

try to derive everything important physics from the basics, every possible connection you can think of, thats good revision, i am trying to do that.

My suggestion is try to have an overview that way you wouldnt get demoralized or stuck by the nitty gritty stuff. So have an overview, try to understand the connections between the concepts, will save you a lot of time, maybe years. With loads of problems ofcourse.

Having said that, I am not very sure about giving advices, if it doesnt work for you or distracts you from your way of approach, that would not be good.

So just try to have an overview.proper revision, ability to derive . :thumbup:

twistor59 · #4 by **twistor59** » Jan 16, 2012 2:34 pm

Here's all the coordinate free proofs in one nice document:

http://ebooksgo.org/mathematics/Vector_Identity.pdf

if you still have trouble with the epsilon ijk versions, I'll have a look tonight if no one else responds.

Joe09 · #5 by **Joe09** » Jan 16, 2012 5:11 pm

thankyou guys, my exam in this was this morning but i appreciate he help nonetheless

the identities did not come up in the exam but i went through them with a friend for the booklet submittion and was able to figure them out

Joe09 · #6 by **Joe09** » Jan 16, 2012 5:12 pm

that pdf is beautiful, that is saved on my pc now