Second partials chain rule

  • Thread starter jbusc
  • Start date
  • #1
212
0
This is a stupid question but...

The regular multivariable chain rule is:

[tex]u_x = u_v v_x + u_w w_x[/tex] and [tex]u_y = u_v v_y + u_w w_y[/tex] where [tex]u(v(x, y), w(x, y))[/tex]

Now, are there formulae for the second partials [tex]u_{xx}, u_{xy}, u_{yy}[/tex]

I just want to check myself (this isn't a homework problem, though it is study for a class) Thanks
 
Last edited:

Answers and Replies

  • #2
AKG
Science Advisor
Homework Helper
2,565
4
Easy, just use product rule:

[tex]u_{xx} = (u_x)_x = (u_vv_x + u_ww_x)_x = u_{vx}v_x + u_vv_{xx} + u_{wx}w_x + u_ww_{xx}[/tex]
 
  • #3
Office_Shredder
Staff Emeritus
Science Advisor
Gold Member
3,907
181
AKG said:
Easy, just use product rule:

[tex]u_{xx} = (u_x)_x = (u_vv_x + u_ww_x)_x = u_{vx}v_x + u_vv_{xx} + u_{wx}w_x + u_ww_{xx}[/tex]
If I recall correctly though, [tex]u_{vx}[/tex] is really [tex]u_{vv}v_x + u_{vw}w_x[/tex]
 
  • #4
AKG
Science Advisor
Homework Helper
2,565
4
Yeah, it probably is.
 
  • #5
212
0
Hmm, yeah. I knew it was straightforward, but it seemed too simple for some reason.
 

Related Threads on Second partials chain rule

  • Last Post
Replies
0
Views
1K
Replies
1
Views
6K
Replies
2
Views
6K
  • Last Post
Replies
2
Views
1K
  • Last Post
Replies
4
Views
11K
  • Last Post
Replies
2
Views
3K
  • Last Post
Replies
7
Views
2K
  • Last Post
Replies
17
Views
2K
  • Last Post
Replies
2
Views
783
  • Last Post
Replies
3
Views
1K
Top