Chapter 4 The Potential Equation 257
From these equations we easily find that the Laplacian in polar coordinates is
∇^2 v=∂(^2) v
∂r^2 +
1
r∂v
∂r+1
r^2∂^2 v
∂θ^2 =1
r∂
∂r(
r∂v
∂r)
+
1
r^2∂^2 v
∂θ^2.In cylindrical(r,θ,z)coordinates, the Laplacian is
∇^2 v=1
r∂
∂r(
r∂v
∂r)
+
1
r^2∂^2 v
∂θ^2 +∂^2 v
∂z^2.EXERCISES
- Find a relation among the coefficients of the polynomial
p(x,y)=a+bx+cy+dx^2 +exy+fy^2that makes it satisfy the potential equation. Choose a specific polynomial
that satisfies the equation, and show that, if∂p/∂xand∂p/∂yare both zero
at some point, the surface there is saddle shaped.- Show thatu(x,y)=x^2 −y^2 andu(x,y)=xyare solutions of Laplace’s equa-
tion. Sketch the surfacesz=u(x,y). What boundary conditions do these
functionsfulfillonthelinesx=0,x=a,y=0,y=b? - If a solution of the potential equation in the square 0<x<1, 0<y< 1
has the formu(x,y)=Y(y)sin(πx), of what form is the functionY?Finda
functionYthat makesu(x,y)satisfy the boundary conditionsu(x, 0 )=0,
u(x, 1 )=sin(πx). - Find a functionu(x), independent ofy, that satisfies the potential equation.
- What functionsv(r), independent ofθ, satisfy the potential equation in
polar coordinates? - Show thatrnsin(nθ)andrncos(nθ)both satisfy the potential equation in
polar coordinates(n= 0 , 1 , 2 ,...). - Find expressions for the partial derivatives ofuwith respect toxandyin
terms of derivatives ofvwith respect torandθ. - Ifuandvare thex-andy-components of the velocity in a fluid, it can be
shown (under certain assumptions) that these functions satisfy the equa-
tions
∂u
∂x+
∂v
∂y=^0 , (A)
∂u
∂x−∂v
∂x=^0. (B)