Question

Need to check whether the problems are correct or not. Topic is tensor analysis

Answer 1

@Michele_Laino

Answer 2

@lochana

Answer 3

@Michele_Laino I think the problems are wrong. but i am not sure

Answer 4

question #1
I start from this equation:
\[\Large {A^k} = \frac{{\partial {x^k}}}{{\partial x{'^i}}}A{'^i}\]

Answer 5

ok.. then

Answer 6

subsequently, I multiply both sides of that equation, by:
\[\Large \frac{{\partial x{'^l}}}{{\partial {x^k}}}\]
so, I get:

Answer 7

\[\Large \frac{{\partial x{'^l}}}{{\partial {x^k}}}{A^k} = \frac{{\partial {x^k}}}{{\partial x{'^i}}}\frac{{\partial x{'^l}}}{{\partial {x^k}}}A{'^i}\]
Please note that sum is implicit ove repeated indices.
After a simplification, I write:
\[\Large \frac{{\partial x{'^l}}}{{\partial {x^k}}}{A^k} = \frac{{\partial {x^k}}}{{\partial x{'^i}}}\frac{{\partial x{'^l}}}{{\partial {x^k}}}A{'^i} = \delta _i^lA{'^i}\]
therefore, I get:
\[\Large \frac{{\partial x{'^l}}}{{\partial {x^k}}}{A^k} = A{'^l}\]

Answer 8

the question is wrong I think, i mean the subscripts will be superscript

Answer 9

subsequently, I note that:
\[\Large A{'^i} = {g_{ik}}A{'^k} = {g_{ik}}\frac{{\partial x{'^k}}}{{\partial {x^m}}}{A^m}\]
that is what all can I write, so your result is wrong :(

Answer 10

So the question has an error in the subscript, right?

Answer 11

oops.. I made a typo:
\[\Large A{'_i} = {g_{ik}}A{'^k} = {g_{ik}}\frac{{\partial x{'^k}}}{{\partial {x^m}}}{A^m}\]

Answer 12

yes! right!

Answer 13

here is the right identity:
\[\Large A{'_i} = \frac{{\partial {x^k}}}{{\partial x{'_i}}}{A_k}\]

Answer 14

which is a definition

Answer 15

Yes, the law of transformation of covariant tensors

Answer 16

correct!

Answer 17

I want to know, is there any relation connecting a covariant tensor and a contravariant tensor?

Answer 18

I think that there is not a relation which allows us to go from covariant tensor to contravariant tensor and vice versa

Answer 19

ok... then the second problem is also wrong I guess.

Answer 20

I try to solve it, please wait...

Answer 21

your result is wrong, since, I can write these steps:
\[\Large \begin{gathered}
A'_i^ju{'^i}v{'_j} = \frac{{\partial x{'^j}}}{{\partial {x^m}}}\frac{{\partial {x^n}}}{{\partial x{'^i}}}A_n^m\frac{{\partial x{'^i}}}{{\partial {x^l}}}{u^l}\frac{{\partial {x^p}}}{{\partial x{'^j}}}{v_p} = \hfill \\
\hfill \\
= \delta _m^p\delta _l^nA_n^m{u^l}{v_p} = A_l^p{u^l}{v_p} \hfill \\
\end{gathered} \]

Answer 22

please wait:

Answer 23

please see the picture above for the second question

Answer 24

Yeah Thanks.. but replacing j by i in
\[A _{j}^{i} \]

Answer 25

makes the problem correct.

Answer 26

so both the problems have typing errors

Answer 27

yes!

Answer 28

thanks... :)

Answer 29

:)