# Tensors - indices

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

## Answers and Replies

cristo
Staff Emeritus
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
Homework Helper
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.

Hi

Thanks for ur response:)

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

Thanks again

MAry:)

#### Attachments

• TENSOR.doc
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tiny-tim
Homework Helper
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