Volume of parametrized manifold

  • #1
I don't think this is a difficult problem, but I am not sure about what is being asked in the question. I got it from Munkres' Analysis on Manifolds page 193 Q 2.


Homework Statement



Let A be open in R^k; let f : A-->R be of class C^r; let Y be the graph of f in R^(k+1), parametrized by the function @(x) = (x, f(x)). Express V([email protected]) as an integral. (this is the volume of the graph of f)


Homework Equations



the volume of a parametrized manifold, or V([email protected]) = integral over A of det(([email protected])^tr [email protected]), where [email protected] is the derivative matrix of @ and [email protected]^tr is the transpose of [email protected]

The Attempt at a Solution



I think I would just use the above equation to express V([email protected]) as an integral. But then I would be done a little too quickly, so I thought that maybe I am supposed to simplify the matrix multiplication or something. Am I missing something here?

Also, can someone tell me how to post using mathematical symbols?
 

Answers and Replies

  • #2
Can anyone confirm I am on the right track or am I way off? Or, at the very least, can someone tell me how to post using mathematical symbols. I am not very computer savvy. :)
 

Related Threads on Volume of parametrized manifold

Replies
1
Views
901
  • Last Post
Replies
2
Views
576
  • Last Post
Replies
8
Views
3K
  • Last Post
Replies
2
Views
1K
  • Last Post
Replies
5
Views
9K
  • Last Post
Replies
4
Views
831
  • Last Post
Replies
4
Views
2K
  • Last Post
Replies
2
Views
1K
  • Last Post
Replies
4
Views
751
  • Last Post
Replies
16
Views
3K
Top