Possible Error in Vladimir Arnold Mathematical Methods C

[xdrgnh]

Main Question or Discussion Point

I'm reviewing differential forms again and I believe there is an error in Vladimir Arnold's Mathematical Methods of Classical Mechanics. On page 176 of the hardcover edition of it, in chapter 7, on the section of differential forms, problem 4 he says that dr^2(r^2=x^2+y^2) acting on the 2nd vector in the problem equals -8, I get that is equal 8. I can't understand why there would be a minus sign considering this dr^2, shouldn't it being raised to the 2nd power force the answer to being negative. If anyone can confirm this is not an error I'll be on my way to figuring it out . Thanks

 

[Orodruin]

You need to quote everything that is relevant to your question. You cannot just say page x in y and hope that someone is sitting with their copy of Arnold right in front of them.

 

[martinbn]

It is -8, the vector $\xi_2$ on the picture is equal to $-\frac{\partial}{\partial x_1}-\frac{\partial}{\partial x_2}$.

 

[mathwonk]

to see this, use that d(r^2) = 2x1dx1 + 2x2dx2, and note that both x1 and x2 are positive on tht vector, but dx1 and dx2 are negative.

 

[xdrgnh]

Thank you all very much, I was able to figure out the correct answer thanks to all of you and learn a bit more about differential forms.