Cauchy-Riemann: Is the Relation ∂x/∂y = -∂y/∂x True in General?

[sahil_time]

Considering f(z) = z where z is analytic. z = x + iy.
f(z) = u + iv = x + iy.
Hence u=x and v=y.
Using Cauchy Reimann eqns.
∂u/∂x = ∂v/∂y =1 and
∂u/∂y = -∂v/∂x where u=x and v=y hence

∂x/∂y = -∂y/∂x

is this relation true in general?

 

[osnarf]

I'm assuming you mean z is some function z(w) of another complex number w. So Yes, since f(z) = z = z(w) is just the identity function.

 

[sahil_time]

I'm assuming you mean z is some function z(w) of another complex number w. So Yes, since f(z) = z = z(w) is just the identity function.

I meant to ask whether the relation is true in general and not just complex numbers.

eg: dy/dx = 1/(dx/dy) for 2 Dimensional plane. Just that way is ∂x/∂y = -∂y/∂x
true for any 2 Dimensional plane?


Thankyou. :)