Can someone help me solve these problems

[shankarvn]

Hi everyone

I am a mechanical engg graduate. Can someone help me solve these problems?
This are problems from our textbook on differential forms by manfredo P Do carmo. I have attached the file.
Thanks

 

[Reb]

Hi everyone

I am a mechanical engg graduate. Can someone help me solve these problems?
This are problems from our textbook on differential forms by manfredo P Do carmo. I have attached the file.
Thanks

For 1, note that the orientation on each chart has to agree on the connected intersection, so it must be either positive or either negative throughout.

For 2, show that the intersection of the charts is connected (it actually is the equator).

For 3, use the uniqueness theorem for initial value problems of differential equations.