Question

In the solution to problem 3 in session 42 the set z =0. My understanding is that dz = 0 not z. Can anyone help explain?

edit:
they only set some of the "z's" in the equation to 0 and leave the rest. Now I'm totaly confused.

Answer 1

I agree, they have no business setting z=0. you have
the total differential
dw = Wx dx + Wy dy + Wz dz

and from the constraint we have dy = g(x,y) dx where g(x,y) is a function of x and y
and so
dw = Wx dx + Wy g(x,y) dx + Wz dz
dw = (Wx + Wy g(x,y) ) dx + Wz dz

and by definition the partial w with respect to x is the expression multiplying the dx:
\[ \frac{\partial w}{\partial x} = W_x + g(x,y) \cdot W_y \]

No constraint on variable z is given, so it should remain (not be set to 0 or any other value)