# Multivariate Chain Rule

1. Jul 21, 2010

### Kreizhn

1. The problem statement, all variables and given/known data
Find
$$\frac{\partial z}{\partial y} [/itex] where z=F(u,v,y), u=f(v,x), v=g(x,y). 3. The attempt at a solution If I remember multivariate calculus at all, this should be (please forgive the abuse of notation) [tex] \frac{\partial z}{\partial y} = \frac{\partial z}{\partial u}\frac{\partial u}{\partial v}\frac{\partial v}{\partial y} + \frac{\partial z}{\partial v}\frac{\partial v}{\partial y} + \frac{\partial z}{\partial y}$$

However, I have a book in front of me that says it should be

$$\frac{\partial z}{\partial y} = \frac{\partial z}{\partial v}\frac{\partial v}{\partial u}\frac{\partial u}{\partial y} + \frac{\partial z}{\partial u}\frac{\partial u}{\partial y} + \frac{\partial z}{\partial y}$$

I think I'm correct and that they must have confused u and v. If someone could verify this for me it would be much appreciated.

2. Jul 21, 2010

### Dick

I agree. It think they mixed up u and v.