# 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

Related Advanced Physics Homework Help News on Phys.org
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!

Hi

Thanks for ur response:)

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

Thanks again

MAry:)

#### Attachments

• 291.5 KB Views: 223
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