Answer to Question #235090 in Chemical Engineering for Loki

Question

8)Show that u(x, y) = 4y + 3x²y-y³ is harmonic and find its conjugate harmonic function v(x, y).

Answer 1

We can show this using the basic, foundational re-expressed equation:

With this we get:

∂u ∂x = 3x 2 y − y 3, ∂ 2u ∂x2 = 6xy and ∂u ∂y = x 3 − 3xy2 , ∂ 2u ∂y2 = −6xy

u = 2x 2 − 2y 2 + 6xy + 4x + 6y and v = −3x 2 + 3y 2 + 4xy + 4y − 6x. ∂u ∂x = 4x + 6y + 4, ∂v ∂y = 6y + 4x + 4 and ∂u ∂y = −4y + 6x + 6, ∂v ∂x = −6x + 4y − 6.

Also shown below: