For the second example that it gives for the chain rule with z=F(u,v,y) where u=f(v,x) & v=g(x,y) I don't really understand the diagram that is drawn & when I got a different answer then the one given by the book for the derivative of z with respect to y.

Since it tells you to think of each arrow meaning depends on I only drew an arrow if the function used one of those variables so I had the following

z with arrows to u,v and y

u with arrows to v and x

v with arrows to x and y

and came up with the following

dz/dy=dz/dy+dz/dv dv/dy+ dz/du du/dv dv/dy

could someone explain to me where i went wrong because I feel like at the very least the diagram has to be wrong because it gives five paths to y and only three are used.