Is the Multivariate Chain Rule Being Applied Correctly?

[Kreizhn]

Homework Statement

Find
[tex]\frac{\partial z}{\partial y} [/itex]<br /> where z=F(u,v,y), u=f(v,x), v=g(x,y).<br /> <br /> <h2>The Attempt at a Solution</h2><br /> If I remember multivariate calculus at all, this should be (please forgive the abuse of notation)<br /> <br /> $$\frac{\partial z}{\partial y} = \frac{\partial z}{\partial u}\frac{\partial u}{\partial v}\frac{\partial v}{\partial y} + \frac{\partial z}{\partial v}\frac{\partial v}{\partial y} + \frac{\partial z}{\partial y}$$<br /> <br /> However, I have a book in front of me that says it should be<br /> <br /> $$\frac{\partial z}{\partial y} = \frac{\partial z}{\partial v}\frac{\partial v}{\partial u}\frac{\partial u}{\partial y} + \frac{\partial z}{\partial u}\frac{\partial u}{\partial y} + \frac{\partial z}{\partial y}$$<br /> <br /> 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.[/tex]

 

[Dick]

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