- #1
LagrangeEuler
- 717
- 20
If we consider function ##z=z(x,y)## then ##dz=(\frac{\partial z}{\partial x})_ydx+(\frac{\partial z}{\partial y})_xdy##. If ##z=const## then ##dz=0##. So,
[tex](\frac{\partial z}{\partial x})_ydx+(\frac{\partial z}{\partial y})_xdy=0[/tex]
and from that
[tex]\frac{dx}{dy}=-\frac{(\frac{\partial z}{\partial y})_x}{(\frac{\partial z}{\partial x})_y}[/tex]
or
[tex](\frac{\partial x}{\partial y})_z=-\frac{(\frac{\partial z}{\partial y})_x}{(\frac{\partial z}{\partial x})_y}[/tex]
My question is is ##z=const## isn't then also ##(\frac{\partial z}{\partial x})_y=0## and ##(\frac{\partial z}{\partial y})_x=0##? This is confusing to me. Could you please answer me that.
[tex](\frac{\partial z}{\partial x})_ydx+(\frac{\partial z}{\partial y})_xdy=0[/tex]
and from that
[tex]\frac{dx}{dy}=-\frac{(\frac{\partial z}{\partial y})_x}{(\frac{\partial z}{\partial x})_y}[/tex]
or
[tex](\frac{\partial x}{\partial y})_z=-\frac{(\frac{\partial z}{\partial y})_x}{(\frac{\partial z}{\partial x})_y}[/tex]
My question is is ##z=const## isn't then also ##(\frac{\partial z}{\partial x})_y=0## and ##(\frac{\partial z}{\partial y})_x=0##? This is confusing to me. Could you please answer me that.