Exploring the Second Partials Chain Rule in Multivariable Calculus

[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.