If U =U(x, y ) is converted to polar coordinates to become U =U(r, θ) one can
treat r=r(x, y )and θ=θ(x, y )to calculate the following derivatives of Uwith respect
xand y
∂U
∂x=∂U
∂r∂r
∂x+∂U
∂θ∂θ
∂x
∂^2 U
∂x^2 =∂U
∂r∂^2 r
∂x^2 +∂r
∂x[
∂^2 U
∂r^2∂r
∂x +∂^2 U
∂r ∂θ∂θ
∂x]+∂U
∂θ∂^2 θ
∂x^2 +∂θ
∂x[
∂^2 U
∂θ ∂r∂r
∂x +∂^2 U
∂θ^2∂θ
∂x](12 .57)and
∂U
∂y=∂U
∂r∂r
∂y+∂U
∂θ∂θ
∂y
∂^2 U
∂y^2 =∂U
∂r∂^2 r
∂y^2 +∂r
∂x[
∂^2 U
∂r^2∂r
∂y +∂^2 U
∂r ∂θ∂θ
∂y]+∂U
∂θ∂^2 θ
∂y^2 +∂θ
∂y[
∂^2 U
∂θ ∂r∂r
∂y +∂^2 U
∂θ^2∂θ
∂y](12 .58)The transformation equations from rectangular to polar coordinates is performed
using the transformation equations
x=rcos θ and y=rsin θ (12 .59)
One can solve for rand θin terms of xand y to obtain
r^2 =x^2 +y^2 and tan θ=
y
x(12 .60)Differentiate the equations (12.60) with respect to xand show
2 r∂r∂x = 2x and sec
(^2) θ∂θ
∂x =−
y
x^2 (12 .61)
which simplifies using the transformation equations (12.59) to the values
∂r
∂x= cos θ and ∂θ
∂x=−sin θ
r(12 .62)Differentiate the the equations (12.60) with respect to yand show
2 r∂r
∂y= 2y and sec^2 θ∂θ
∂y=^1
x(12 .63)Use the transformation equations (12.59) and simplify the equations (12.63) and
show
∂r∂y = sin θ and
∂θ
∂y =cos θ
r (12 .64)