Index Notation - Vector Calculus Help

So I know vaguely how to do it for some basic examples but wanted to know why some of it worked and what are the general rules so i don't get stuck on harder or unfamiliar questions

so when i prove a x (bxc) = b(a.c)-c(a.b) i get to the answer but wanted to know/understand what things are allowed:

after the first few steps you get to

Eijk aj Eklm bl cm

on the next step can you now move the second Eklm to the front, why? because its a constant??...

and also is the following equivalent?

EijkEklm aj bl cm =EijkEklm bl cm aj = EijkEklm cm bl aj

if so why? and can you always interchange the vectors like this

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Original post by Jammy4410

You have to remember that these aren't vector equations. Everything in these equations are components of vectors (or other tensors) and are, as such, scalars. So yes, you can rearrange their order at will.

All the important vector-y-ness is encoded in the indices, so as long as you maintain those relationships you're fine. Remember, they show the order in which you sum over components. In an ordinary vector equation this is shown by ordering vectors and matrices in a specific way, which is where your confusion comes in.

So yes, the final is equivalent.

(edited 9 years ago)

Reply 3

Jammy4410

Original post by Rinsed

Ok thanks thats cleared some things up but I've got 5-6 problems I'm still confused by. So il do them one by one

Reply 4

Jammy4410

First one is:

a x a

( a x a)i (ith component)

= Eijk aj ak
= -Eikj aj ak (Swapping order in E means It will be minus, so an extra minus sign is needed for equality)

= -Eikj ak aj. (Components of vectors, I.e scalar so can interchange)

= - (a x a)i

And since (a x a)i = -(a x a)i then a x a = 0

Is this right?

Reply 5

Jammy4410

I just realised the above proof must be wrong. But why?

I know this cause I could do the same thing with

a x b where a does not equal b and it would imply a x b = 0

Reply 6

MindTheGaps

Original post by Jammy4410

Yep, looks good to me.

Original post by Jammy4410

I just realised the above proof must be wrong. But why?

I know this cause I could do the same thing with

a x b where a does not equal b and it would imply a x b = 0

Nope, it tells you that a x b= -b x a, which is fine. It's only when the vectors are the same when this swapping doesn't change anything meaning it must be zero. Remember, you swapped the order of the indices, which is equivalent to changing the order of the vector equation.

Reply 7

Jammy4410

Original post by Rinsed

Yep, looks good to me.

Nope, it tells you that a x b= -b x a, which is fine. It's only when the vectors are the same when this swapping doesn't change anything meaning it must be zero. Remember, you swapped the order of the indices, which is equivalent to changing the order of the vector equation.

Oh yeah, I'm being silly.

Problem 2: The divergence of the curl

(Gonna use V for the grad sign, forgotten the name atm, and u for the arbitrary vector) and di for the index notation for V

V . (V x u) (i) Ith component

= di Eijk dj uk
=Eijk dj di uk (orde don't matter)
= -Ejik dj di uk
= -dj (V x u) jth component
= - V. (V x u). Jth company

A property of a scalar is that
a = a transpose

And since this produces a scalar then we have shown

a= a transpose = -a if and only if a=0

Is the reasoning correct?

Reply 8

Jammy4410

Problem 3: involving scalar functions
An arbitrary vector crossed with the grad of a scalar function

(Gonna use a for the scalar function)

Va x u ith component

= Eijk dja uk
= Eijk dj auk

=V x au

I know this is wrong but why? And why cant you move scalars about like this

Is it because you have to treat 'dj a ' together as one thing and cant be separated? An that the way you write it out makes it look like a constant that can be moved around

(edited 9 years ago)

Reply 9

MindTheGaps

Original post by Jammy4410

Tbh I'm unsure why you're transposing things, is that in the question? But your reasoning looks correct, if you're trying to show the divergence of a curl is 0.

Original post by Jammy4410

Yea so, grad a is a vector, of which you are looking at the components. You can't just split up the del operator and the scalar function. The scalar function itself does not exist in your equation, only the jth component of its gradient.

Also you probably want to write it like

( \nabla a )^i

To avoid confusion, rather than putting the index on the del.

Reply 10

Jammy4410

Original post by Rinsed

Tbh I'm unsure why you're transposing things, is that in the question? But your reasoning looks correct, if you're trying to show the divergence of a curl is 0.

Yea so, grad a is a vector, of which you are looking at the components. You can't just split up the del operator and the scalar function. The scalar function itself does not exist in your equation, only the jth component of its gradient.

Also you probably want to write it like

( \nabla a )^i

To avoid confusion, rather than putting the index on the del.

i mentioned the transpose cause in my working i showed that

the ith component of V.(V x u) was equal to the jth component of - V.(V x u)

Reply 11

Jammy4410

problem 4:

how do i show that

V. (aU) = (Va).U only if a is a constant vector

where
V is the del operator
a is a scalar field
U is an arbitrary vector field

Reply 12

MindTheGaps

Original post by Jammy4410

i mentioned the transpose cause in my working i showed that

the ith component of V.(V x u) was equal to the jth component of - V.(V x u)

That's a fully contracted tensor (i.e. a scalar) so ith or jth components or whatever doesn't make sense.

You showed that

\nabla \cdot (\nabla \times \textbf{u})=-\nabla \cdot (\nabla \times \textbf{u})=0

Reply 13

MindTheGaps

Original post by Jammy4410

problem 4:

how do i show that

V. (aU) = (Va).U only if a is a constant vector

where
V is the del operator
a is a scalar field
U is an arbitrary vector field

So I'll set it up for you.

You have, in index notation,

\nabla^{i} (a {U}_{i})

(If you're not familiar with higher and lower indices, just ignore it)

The del is a differential operator, just use the product rule and see what happens.

(edited 9 years ago)

Reply 14

Jammy4410

Original post by Rinsed

So I'll set it up for you.

You have, in index notation,

\nabla^{i} (a {U}_{i})

(If you're not familiar with higher and lower indices, just ignore it)

The del is a differential operator, just use the product rule and see what happens.

i thought you had to use the product rule but was having trouble knowing when to use it and when you can just interchange, i guess this is 'di' of a product unlike the others:

i got:

V. (aU) ith

= di (aU)i
= di (aUi)
=(dia)Ui + a(diUi)
= (V.a) + a(V.U) ith component

is that right?

and if a is constant then the first term is zero.

so we get

V.(aU) = a(V.U) and then I'm guessing you can re write the RHS as (Va).U why??

Reply 15

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??

Reply 16

MindTheGaps

Original post by Jammy4410

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.

Reply 17

Jammy4410

Original post by Rinsed

im a bit confused now..

in what way was it right?
and what way was it wrong? (based on what each letter is defined as)

Reply 18

Jammy4410

i think i get it, if a was a scalar the first term would be automatically zero but with 'a' as a vector the first term will appear but be zero if a is a constant vector