Tensors: Exploring Indices, Equations, & Transformations

[DreamComeTrue]

1. (a) Remembering the distinction between summation indices and free indices, look at the following equations and state whether they conform to tensor notation, and if not why not:
(i) Tmn=Am^nB
(ii) Uij^i=Ai^kDk
(iii) Vjk^ii=Ajk
(iv) Ai^j=Xi^iC^j+Yi^j

(b) (i) Write out in full the equations bi=djgi^j (d-differential) in a 2-dimensional space.
(ii) If g^ij is the inverse of the metric tensor gij and di=d/dx^i, what are the values of the components of bi=djgi^j ?


(c) For this part, you should use the tensor transformation rules for a contravariant and covariant vector, and for a second-rank contravariant tensor:
P^i=(dx^i/dx^a)P^a, Pi=(dx^a/dx^i)Pa, T^ij=(dx^i/dx^a)(dx^j/dx^b)T^ab
, , .

(i) If A^i and B^j are contravariant vectors, prove that transforms as A^iB^j a contravariant second-rank tensor.



(ii) If A^i is a contravariant vector and Ci is a covariant vector, prove that A^iCi is a scalar field.


(iii) If T^ij is a skew-symmetric contravariant second-rank tensor, prove that its skew symmetry property is invariant under tensor transformations.

Anyone who can help me with that?

Many thanks

Mary

 

[cristo]

Firstly, we don't do people's homework for them here. You must show your work before we can help you.

Secondly, your notation is ambiguous: it is common to denote subscripts with underscores (A_n), superscripts with hats, or whatever they're called (A^n), and multiple indices in brackets (A^{mn}).

 

[tiny-tim]

Welcome to PF!

Hi Mary! Welcome to PF!

Please write your questions again, using the X² and X₂ tags just above the Reply box … they are almost ureadable now.

 

[DreamComeTrue]

Hi

Thanks for ur response:)

I attached the relevant file, hope that now it's easier

Thanks again

MAry:)

 

[tiny-tim]

Hi Mary! Thanks for the PM.

Sorry, I don't like .doc files …

I was hoping someone else would anwser …

can't you type it?

 

[nicksauce]

While the .doc file is certainly easier to read...
1) It would be nice for you to re-post here, as many people don't like .doc files, and as Tiny-Tim said there are X^2 and X_2 tags in the reply box.
2) As Cristo said, we don't do peoples' homework here. Sure we can help you with it... but please post an attempt at the problem, or indicate specifically what is giving you problems, so that we may have an easier time helping without doing all of your work for you.

First of all, can you tell us what conditions need to be satisfied in a proper tensor algebraic expression?