A Pullback on a manifold

  • Thread starter Rico1990
  • Start date
3
0
Hey,
we had in the lecures the following:
Let M and N be smooth manifolds, and dim(M)=dim(N)=n, while $$x^i$$ and $$ y^i$$ are coordinate functions around $$p\in M$$ respective $$F(p) \in N$$, then we get for the pullback of F
Untitled01.jpg

Which entries has the matrix we take the determinant of? I thaught of partial derivatives but am not sure.
 

Attachments

Orodruin

Staff Emeritus
Science Advisor
Homework Helper
Insights Author
Gold Member
2018 Award
15,844
5,842
Yes it is the partial derivatives of ##y^j \circ F## with respect to ##x^i##, which is what the image you pasted says. This is just the Jacobian of the transformation ##x^i \to y^j## as subsets of ##\mathbb R^n##.
 
3
0
Ok, thank you for your answer. But answer me please two last questions that arose. I deduce that these partial derivatives are defined, but they are vague in the sense, that $$x^i , y^i$$ are both functions the derivatives depend on. Is it the interest to leave them this vague or does one insert certain values so that the coordinate functions give „real" coordinates.
What is the use of this formula?

Best wishes Rico
 

Want to reply to this thread?

"Pullback on a manifold" You must log in or register to reply here.

Related Threads for: Pullback on a manifold

  • Posted
Replies
4
Views
1K
Replies
5
Views
4K
Replies
8
Views
7K
  • Posted
Replies
0
Views
4K
Replies
1
Views
1K
  • Posted
Replies
6
Views
4K
  • Posted
Replies
17
Views
954
Replies
9
Views
595

Physics Forums Values

We Value Quality
• Topics based on mainstream science
• Proper English grammar and spelling
We Value Civility
• Positive and compassionate attitudes
• Patience while debating
We Value Productivity
• Disciplined to remain on-topic
• Recognition of own weaknesses
• Solo and co-op problem solving
Top