Derivative of a Tensor: Solving for d_a A^b

[SunGod87]

Homework Statement

http://img522.imageshack.us/img522/3511/80377551yt7.jpg

Homework Equations

None... I think. Seems like something I should just know rather than have to work out?

The Attempt at a Solution

I can do everything in this problem apart from the very first part. I thought to work out how d_a A^b transforms I'd have to start from d'_a A'^b but this must not be the case as it's the next part of the question?

 

[Dick]

Use that A^b is a tensor to figure out what A'^b looks like. Then use the chain rule for partial derivatives to figure out how d'_a A'^b is related to d_a A^b. You should wind up with an term that doesn't belong if d_a A^b actually were a tensor.

 

[SunGod87]

Sure, I've done that for the next part: "compute d'_a A'^b" and found it isn't a tensor so the partial derivative operator isn't a good operator in tensor analysis and a good operator should return a tensor. But what about the first part? I can't answer it without having attempted the next parts to the question...

 

[HallsofIvy]

What first part? Just saying how it transforms under a change of coordinates? Yes, you will have to determine what the derivative is. Fortunately, you say you have done that.

 

[SunGod87]

Thing is, I've asked my lecturer about this now. He says we should answer that part first? Surely he's either made a typo or he's wrong? You HAVE to do the second part to know the answer to the first, right?

In fact, while I have your attention... can you check my working, please?
http://img523.imageshack.us/img523/641/39518542gl8.jpg

 

[Dick]

What second part do you have to do first? You start by finding how the expression tranforms. The expression you sent in is roughly ok, but it's pretty confusing because you are using the same symbols 'a' and 'b' for the dummy indices you are supposed to be summing over. Chose different symbols for those.

 

[SunGod87]

My question is, I can only answer "How does d_a A^b transform under coordinate transformations" after having done: "compute d'_a A'^b", right?

I've also revised my solution to the second part here
http://img88.imageshack.us/img88/3954/38821206vp5.jpg
This is what you meant?

Thanks for the help so far!

 

[Dick]

Your indices still aren't matching up quite right. In the first term in your final line after you multiply it out, you have two 'a's, two 'd's, one 'b' and one 'c'. That's not right. I actually think of "How does d_a A^b transform under coordinate transformations" and "compute d'_a A'^b" as the same question. Sure, you have to do the computation first to see how it transforms.

 

[SunGod87]

That's what I thought, but the lecturer is saying otherwise. I'm tempted to think he just misunderstood my question, since "How does d_a A^b transform under coordinate transformations" doesn't actually have a question mark after it, maybe it's just a sentance to describe the rest of the question.

http://img206.imageshack.us/img206/5706/96320057by8.jpg Noticed the slip, should be okay now?

 

[Dick]

Yeah, I think the "How does" part is the motivation for obeying the command "compute". And, yes, it just looks like a typo.

 

[SunGod87]

Thanks a lot!