# Index Notation - Vector Calculus Help watch

1. (Original post by Jammy4410)
i just realised i did mean that 'a' was a scalar field and U was a constant vector sorry about that
No problem, I was pretty sure you just misread it.

(Original post by Jammy4410)
problem 5 (based on problem 3, similar to problem 4):

is there any similar relationship for the vector product i.e.

is there a relationship between:

Va x U and V x aU, and maybe even equivalent if a is a constant vector??
I don't want to just give you the answer, but write it out in component form, and remember that the components of del have the form , and proceed. It's just product rule and a bit of rearrangement.
2. (Original post by Rinsed)
Not quite.

You actually said a was a constant vector. Because a is a scalar, I assumed you meant U was a constant vector. Worth saying, that gives the right answer. Are you sure the question said a was constant?

Plus, if a were a constant, the fact that the answer includes a differential of a would seem strange.
ok i think i got it:

V.(aU) = (Va).U + a(V.U)

where
V is del
a is scalar field

1) if U is a constant v.field the second term becomes zero
3. (Original post by Rinsed)
No problem, I was pretty sure you just misread it.

I don't want to just give you the answer, but write it out in component form, and remember that the components of del have the form , and proceed. It's just product rule and a bit of rearrangement.
ok i got

V x aU = (Va) x U + a(V x U)

where

V is del
a is s.field
U is v.field

i feel like this is wrong..

method:

=Eijk dj (aU)k
=Eijk dj aUk
=Eijk (dja)Uk + Eijk a(djUk)
=Eijk (dja)Uk + Eijk a(djUk)
= (Va) x U + a(V x U)
4. also, for problem 4:

would there be anyway to get to the answer starting with

(Va).U

if there is a direct method, i think i still don't understand a little of what i can and cannot do
5. (Original post by Jammy4410)
ok i got

V x aU = (Va) x U + a(V x U)

where

V is del
a is s.field
U is v.field

i feel like this is wrong..

method:

=Eijk dj (aU)k
=Eijk dj aUk
=Eijk (dja)Uk + Eijk a(djUk)
=Eijk (dja)Uk + Eijk a(djUk)
= (Va) x U + a(V x U)
Nah I'm pretty sure that's right.

But don't forget that if U is a constant its curl goes away.
6. (Original post by Jammy4410)
also, for problem 4:

would there be anyway to get to the answer starting with

(Va).U

if there is a direct method, i think i still don't understand a little of what i can and cannot do
Well, yes, just apply the steps in reverse. In component form you have

Noting that U is a constant, it can be brought into the bracket. Then it's pretty obvious you can write it as

Remember, the same rules apply here as for any old scalar maths (as long as you don't start messing around with indices) and del is just a differential operator, like d/dx.
7. (Original post by Rinsed)
Well, yes, just apply the steps in reverse. In component form you have

Noting that U is a constant, it can be brought into the bracket. Then it's pretty obvious you can write it as

Remember, the same rules apply here as for any old scalar maths (as long as you don't start messing around with indices) and del is just a differential operator, like d/dx.
still a bit confused how to do it in reverse?

problem 6:

we know that

a x (b x c) = b(a.c) -c(a.b)

why is this not the case for

V x (bxc)

where a,b,c arbitrary vectors

il check it in an hour, but i think i know the answer and how to prove it using index notation, just don't really get why it works

on page 5 of:
http://internal.physics.uwa.edu.au/~...g/CM/index.pdf

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