What are the Second Partials in the Multivariable Chain Rule?

[jbusc]

This is a stupid question but...

The regular multivariable chain rule is:

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

Now, are there formulae for the second partials u_{xx}, u_{xy}, u_{yy}

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

 

[AKG]

Easy, just use product rule:

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}

 

[Office_Shredder]

Easy, just use product rule:

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}

If I recall correctly though, u_{vx} is really u_{vv}v_x + u_{vw}w_x

 

[AKG]

Yeah, it probably is.

 

[jbusc]

Hmm, yeah. I knew it was straightforward, but it seemed too simple for some reason.