Let $y = f(x)$ and $z = g(x, y) $ be real functions of one variable $x$ and two variables $(x,y)$ respectively. Suppose $$dz \wedge dx =0.$$ What conclusion can be drawn from this statements?
My attempt: $$0 = dz\wedge dx =(\frac{\partial g}{\partial x} dx+ \frac{\partial g}{\partial y}dy)\wedge dx = \frac{\partial g}{\partial y}dy\wedge dx. $$ Can we conclude $\frac{\partial g}{\partial y}=0$ or the above statement is always true (because of that $\frac{\partial g}{\partial y}dy\wedge dx = \frac{\partial g}{\partial y} \frac{\partial f}{ dx} dx\wedge dx$ and $dx\wedge dx =0$)?
Thank you for any help.