Multivariable calculus question

  • Thread starter evilcman
  • Start date
  • #1
41
2
I have a scalar function f dependent on a few variables $x_i$, and I would like to change variables, so that [tex]y_i = \sum_j {M_{ij} x_j},[/tex] where M is an invertible matrix independent of the x_i-s, and compute:
[tex]
\frac{\partial f}{\partial x_i} = \frac{\partial f}{\partial \left( \sum_j {M^{-1}_{ij} y_j} \right)} = \sum_j Q_{ij} \frac{\partial f}{\partial y_j}
[/tex]

I suggest that the last identity is true for some matrix Q. Is there a general formula for Q?
 

Answers and Replies

  • #2
lurflurf
Homework Helper
2,440
138
by the chain rule
[tex]
\frac{\partial f}{\partial x_i} = \sum_j \frac{\partial y_j}{\partial x_i} \frac{\partial f}{\partial y_j}= \sum_j Q_{ij} \frac{\partial f}{\partial y_j}
[/tex]
so
[tex]Q_{ij}=\frac{\partial y_j}{\partial x_i}[/tex]

"I suggest that the last identity is true for some matrix Q. Is there a general formula for Q?"

It is, use the chain rule to compute it. It is obvious.
 
Last edited:

Related Threads on Multivariable calculus question

  • Last Post
Replies
3
Views
4K
Replies
3
Views
1K
Replies
1
Views
2K
  • Last Post
Replies
11
Views
920
  • Last Post
Replies
2
Views
2K
  • Last Post
Replies
3
Views
2K
  • Last Post
Replies
1
Views
5K
  • Last Post
Replies
16
Views
3K
  • Last Post
Replies
5
Views
11K
  • Last Post
Replies
11
Views
364
Top