The definition of the stream function in vector notation is

The definition of the stream function in vector notation is u = z times nabla psi. Verify that this definition is equivalent to: partial differential u_x/partial differential x + partial differential u_y/partial differential y = 0

Solution

Taking unit vectors in x, y and z directions as i, j, k we get

u = z X grad psi

u = k X [(del psi / delx)i + (del psi / dely)j + (del psi /delz)k]

u = (del psi / delx)j - (del psi / dely)i

Comparing it with u = u_x i + u_y j we get,

u_x = -(del psi / dely)

u_y = (del psi / delx)

Taking derivatives we get,

del u_x / delx = -del2 psi / (delx dely)

del u_y / dely = del2 psi / (delx dely)

Adding both we get,

del u_x / delx + del u_y / dely = 0..........Hence proved.